PROBLEM INDEX - freewebs.com · 'MATHCOUNTS 2001Œ02 1 2001-2002 MATHCOUNTS School Handbook...

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'MATHCOUNTS 2001-02 PROBLEM INDEX Algebraic Expressions & Equations WU 2-8 WU 3-2 WU 3-6 WU 4-3 WU 9-10 WU 10-3 WU 11-6 WU 12-3 WU 15-1 WU 15-3 WU 16-1 WU 17-3 WO 1-1 WO 1-2 WO 1-6 WO 2-4 WO 3-1 WO 3-5 WO 6-3 WO 6-6 WO 8-4 WO 8-5 WO 8-6 WO 8-7 Coordinate Geometry WU 10-6 WU 12-6 WU 13-9 WU 14-9 WU 16-6 WU 18-6 WU 18-7 WO 4-7 Counting & Combinatorics WU 2-3 WU 5-4 WU 6-5 WU 6-10 WU 10-9 WU 12-5 WU 12-8 WU 13-2 WU 14-5 WU 16-4 WO 3-9 WO 5-2 WO 6-10 WO 9-8 General Math WU 1-1 WU 2-1 WU 2-5 WU 3-8 WU 3-9 WU 4-2 WU 5-5 WU 7-1 WU 7-10 WU 8-1 WU 8-2 WU 9-2 WU 10-1 WU 11-3 WU 12-1 WU 12-2 WU 13-1 WU 13-3 WU 13-5 WU 14-3 WU 17-1 WU 17-2 WU 18-1 WU 18-5 WO 1-3 WO 1-4 WO 1-5 WO 1-10 WO 2-1 WO 2-2 WO 3-2 WO 3-4 WO 3-7 WO 4-4 WO 4-6 WO 6-1 WO 6-7 WO 7-2 WO 7-10 Logic WU 1-9 WU 3-4 WU 5-8 WU 5-9 WU 6-8 WU 7-2 WU 7-6 WU 8-7 WU 9-5 WU 10-2 WU 10-4 WU 11-9 WU 12-7 WU 12-10 WU 13-4 WU 13-8 WU 14-4 WU 14-10 WU 15-10 WU 17-6 WU 18-3 WO 2-9 WO 4-2 WO 5-4 WO 7-7 WO 8-3 WO 8-8 WO 9-9 Miscellaneous Problem Solving WU 2-7 WU 4-1 WU 4-6 WU 6-2 WU 6-7 WU 7-5 WU 9-1 WU 9-9 WU 11-2 WU 11-5 WU 11-10 WU 15-6 WU 15-9 WU 16-2 WU 16-3 WU 18-2 WU 18-10 WO 2-6 WO 2-7 WO 4-1 WO 4-8 WO 5-1 WO 6-8 WO 7-1 WO 9-2 Number Theory WU 1-5 WU 1-7 WU 1-8 WU 1-10 WU 2-6 WU 3-5 WU 3-10 WU 4-4 WU 5-1 WU 7-7 WU 7-8 WU 8-6 WU 9-3 WU 10-10 WU 11-4 WU 12-4 WU 13-6 WU 14-8 WU 15-2 WU 15-4 WU 15-5 WU 16-9 WU 16-10 WU 17-8 WU 18-9 WO 1-9 WO 3-10 WO 4-5 WO 5-10 WO 6-2 WO 6-5 WO 7-6 WO 7-9 WO 8-1 WO 8-9 WO 8-10 WO 9-3 Pattern Recognition WU 4-7 WU 6-4 WU 8-3 WU 9-7 WU 10-8 WU 16-5 WU 16-7 WO 2-3 WO 3-3 WO 5-6 WO 7-5 WO 9-5 Plane Geometry WU 1-2 WU 1-3 WU 2-9 WU 3-7 WU 4-8 WU 4-9 WU 4-10 WU 5-2 WU 5-3 WU 6-1 WU 7-3 WU 8-5 WU 8-8 WU 8-9 WU 8-10 WU 9-4 WU 9-6 WU 10-7 WU 11-1 WU 12-9 WU 13-10 WU 14-6 WU 14-7 WU 15-8 WU 17-7 WU 17-10 WU 18-4 WO 1-7 WO 1-8 WO 3-6 WO 3-8 WO 4-9 WO 4-10 WO 5-3 WO 5-5 WO 5-9 WO 6-9 WO 7-4 WO 7-8 WO 8-2 WO 9-1 WO 9-4 WO 9-6 Probability WU 1-6 WU 2-10 WU 4-5 WU 5-6 WU 6-3 WU 7-9 WU 8-4 WU 9-8 WU 10-5 WU 11-8 WU 13-7 WU 15-7 WU 16-8 WO 9-7 Radicals & Exponents WU 5-10 WU 14-2 WU 18-8 WO 2-10 WO 4-3 WO 5-7 WO 5-8 WO 7-3 WO 9-10 Series & Sequences WU 2-4 WU 5-7 WU 17-5 Solid Geometry WU 1-4 WU 3-3 WU 6-6 WU 7-4 WU 17-4 WU 17-9 WO 2-5 WO 2-8 WO 6-4 Statistics WU 2-2 WU 3-1 WU 11-7 Transfomational Geometry WU 6-9 WU 14-1

Transcript of PROBLEM INDEX - freewebs.com · 'MATHCOUNTS 2001Œ02 1 2001-2002 MATHCOUNTS School Handbook...

Page 1: PROBLEM INDEX - freewebs.com · 'MATHCOUNTS 2001Œ02 1 2001-2002 MATHCOUNTS School Handbook STRETCHES The Stretches, created to give Mathletes practice with specific subject areas,

©MATHCOUNTS 2001-02

PROBLEM INDEX

AlgebraicExpressions& Equations

WU 2-8WU 3-2WU 3-6WU 4-3WU 9-10WU 10-3WU 11-6WU 12-3WU 15-1WU 15-3WU 16-1WU 17-3WO 1-1WO 1-2WO 1-6WO 2-4WO 3-1WO 3-5WO 6-3WO 6-6WO 8-4WO 8-5WO 8-6WO 8-7

CoordinateGeometry

WU 10-6WU 12-6WU 13-9WU 14-9WU 16-6WU 18-6WU 18-7WO 4-7

Counting &Combinatorics

WU 2-3WU 5-4WU 6-5WU 6-10WU 10-9WU 12-5WU 12-8WU 13-2WU 14-5WU 16-4WO 3-9WO 5-2

WO 6-10WO 9-8

General Math

WU 1-1WU 2-1WU 2-5WU 3-8WU 3-9WU 4-2WU 5-5WU 7-1WU 7-10WU 8-1WU 8-2WU 9-2WU 10-1WU 11-3WU 12-1WU 12-2WU 13-1WU 13-3WU 13-5WU 14-3WU 17-1WU 17-2WU 18-1WU 18-5WO 1-3WO 1-4WO 1-5WO 1-10WO 2-1WO 2-2WO 3-2WO 3-4WO 3-7WO 4-4WO 4-6WO 6-1WO 6-7WO 7-2WO 7-10

Logic

WU 1-9WU 3-4WU 5-8WU 5-9WU 6-8WU 7-2WU 7-6WU 8-7

WU 9-5WU 10-2WU 10-4WU 11-9WU 12-7WU 12-10WU 13-4WU 13-8WU 14-4WU 14-10WU 15-10WU 17-6WU 18-3WO 2-9WO 4-2WO 5-4WO 7-7WO 8-3WO 8-8WO 9-9

MiscellaneousProblemSolving

WU 2-7WU 4-1WU 4-6WU 6-2WU 6-7WU 7-5WU 9-1WU 9-9WU 11-2WU 11-5WU 11-10WU 15-6WU 15-9WU 16-2WU 16-3WU 18-2WU 18-10WO 2-6WO 2-7WO 4-1WO 4-8WO 5-1WO 6-8WO 7-1WO 9-2

NumberTheory

WU 1-5

WU 1-7WU 1-8WU 1-10WU 2-6WU 3-5WU 3-10WU 4-4WU 5-1WU 7-7WU 7-8WU 8-6WU 9-3WU 10-10WU 11-4WU 12-4WU 13-6WU 14-8WU 15-2WU 15-4WU 15-5WU 16-9WU 16-10WU 17-8WU 18-9WO 1-9WO 3-10WO 4-5WO 5-10WO 6-2WO 6-5WO 7-6WO 7-9WO 8-1WO 8-9WO 8-10WO 9-3

PatternRecognition

WU 4-7WU 6-4WU 8-3WU 9-7WU 10-8WU 16-5WU 16-7WO 2-3WO 3-3WO 5-6WO 7-5WO 9-5

PlaneGeometry

WU 1-2WU 1-3WU 2-9WU 3-7WU 4-8WU 4-9WU 4-10WU 5-2WU 5-3WU 6-1WU 7-3WU 8-5WU 8-8WU 8-9WU 8-10WU 9-4WU 9-6WU 10-7WU 11-1WU 12-9WU 13-10WU 14-6WU 14-7WU 15-8WU 17-7WU 17-10WU 18-4WO 1-7WO 1-8WO 3-6WO 3-8WO 4-9WO 4-10WO 5-3WO 5-5WO 5-9WO 6-9WO 7-4WO 7-8WO 8-2WO 9-1WO 9-4WO 9-6

Probability

WU 1-6WU 2-10WU 4-5WU 5-6WU 6-3

WU 7-9WU 8-4WU 9-8WU 10-5WU 11-8WU 13-7WU 15-7WU 16-8WO 9-7

Radicals &Exponents

WU 5-10WU 14-2WU 18-8WO 2-10WO 4-3WO 5-7WO 5-8WO 7-3WO 9-10

Series &Sequences

WU 2-4WU 5-7WU 17-5

SolidGeometry

WU 1-4WU 3-3WU 6-6WU 7-4WU 17-4WU 17-9WO 2-5WO 2-8WO 6-4

Statistics

WU 2-2WU 3-1WU 11-7

TransfomationalGeometry

WU 6-9WU 14-1

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©MATHCOUNTS 2001�02 1

2001-2002 MATHCOUNTS School Handbook

STRETCHES

The Stretches, created to give Mathletes practice with specific subject areas, focus on the following areas:

Geometry Topics in Plane and Solid Geometry

Algebra Symbolic Manipulation and Algebraic Thinking

Factoring Probability and Number Sense Involving Factors

As part of the MATHCOUNTS coaching phase, the Stretches can be used to prepare Mathletes for moreadvanced problem-solving situations. They can be used prior to the Warm-Ups and Workouts to introducemathematical topics, or they can be used to teach and reinforce concepts after Mathletes have attemptedthe Warm-Ups and Workouts. Finally, they can be used when preparing for competition to aid Mathleteswith troublesome concepts.

Answers to the Stretches include one-letter codes, in parentheses, indicating appropriate problem-solvingstrategies. However, students should be encouraged to find alternative methods of solving the problems;their methods may be better than the one provided! The following strategies are used: C (Compute),F (Formula), M (Model/Diagram), T (Table/Chart/List), G (Guess & Check), S (Simpler Case),E (Eliminate) and P (Patterns).

MATHCOUNTS Symbols and Notation

Standard abbreviations have been used for units of measure. Complete words or symbols are alsoacceptable. Square units or cube units may be expressed as units2 or units3.

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©MATHCOUNTS 2001�02 2

Geometry Stretch1. _______ How many square units are in the greatest area that can be enclosed by a rectangle

whose perimeter is 20 units?

2. _______ What is the number of units in the perimeter of a triangle bound by the x-axis, they-axis and the line y = x + 3 ?

3. _______ Circle E is inscribed in square ABCD. If the length of segment ABis 4 inches, how many square inches are in the area of the shadedregion? Express your answer in terms of .

4. _______ Three angles of a pentagon have measures 88°, 124° and 92°. If the measures ofthe remaining 2 angles are equal, what is the measure, in degrees, of one of theremaining angles?

5. _______ What is the number of inches in the height of an equilateral triangle whoseperimeter is 30 inches? Express your answer in simplest radical form.

6. _______ and are similar right triangles. The two legs of ∆$%& are 5 cm and6 cm in length. If the area of ∆'() is 135 cm2, what is the number of centimetersin the length of the longer leg of ∆'()?

7. _______ What is the number of square units in the area of the regular hexagon ABCDEF ifsegment DE is equal to 4 units? Express your answer in simplest radical form.

8. _______ Segment AB has endpoints at A (-1, 2) and B (3, 1). Segment AB is reflected overthe y-axis such that A becomes A� and B becomes B�. What is the positivedifference between the lengths of segment AA� and segment BB�?

9. _______ If the length of the edge of a cube is increased by 50%, what is the percentincrease in the volume of the cube? Express your answer to the nearest wholenumber.

10. ______ A flag pole is placed in the sand with the top of the flagpole standing 10 feet above the ground. A 26-foot stringis attached to the top of the flag pole at point A. Holdingthe string to the ground, what is the number of squarefeet in the area of the largest circle that can be drawn inthe sand with the end of the string? Express your answerin terms of π .

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©MATHCOUNTS 2001�02 3

Algebra Stretch1. _______ A straight line passes through the three points (3, -4), (5, 1) and (7, y). What is the

value of y ?

2. _______ What is the value of for x = 100?

3. _______ If I = E/R, E = 87 and I = 3, then what is the value of R?

4. _______ What is the value of n if ?

5. _______ The sum of 7 consecutive integers is 413. What is their mean?

6. _______ The sum of two numbers is 5 and their difference is 11. What is the product of thetwo numbers?

7. _______ Point P is the point of intersection of the horizontal line through (4, 2) and thevertical line through (-5, 5). What is the sum of the coordinates of point P?

8. _______ There are only bicycles and tricycles in Tracy�s backyard. She correctly counted atotal of 30 seats and 70 wheels in the backyard. How many tricycles are in herbackyard?

9. _______ Carrie has an 88% average in biology after all four of the markingperiods. If the final exam counts twice as much as each of thefour marking periods, what percent must Carrie make on the finalexam to have a final average of 90% for the course?

10. ______ The temperature t of the air in degrees Fahrenheit is related to the number n ofchirps a cricket makes in a minute by the formula: . How many times perminute does a cricket chirp when the air temperature is 52° F?

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©MATHCOUNTS 2001�02 4

Factoring Stretch1. _______ What is the sum of all of the distinct, positive prime factors of 1260?

2. _______ What is the product of all of the values of n that make 546,324,16n divisible by 6?

3. _______ Billy tosses one fair 6-sided die with faces labeled 1 through 6. He records theoutcome. Billy does this three more times and the product of his four outcomes is120. How many possible combinations of 4 rolls could he have rolled? (Rolling a 1, 1,2, 2 is considered the same combination as rolling a 1, 2, 2, 1.)

4. _______ The numbers 1 - 400, inclusive, are put into a hat. What is the probability that thefirst number chosen at random is a multiple of 4 or 17? Express your answer as acommon fraction.

5. _______ If a and b are distinct, odd primes, then how many distinct positive factors does4a2b3 have?

6. _______ What is the smallest positive integer that has 2, 3, 4, 6, 7 and 12 as factors?

7. _______ What is the sum of the three greatest consecutive integers less than 200 for whichthe least number has 4 as a factor, the second number has 5 as a factor and thegreatest number has 6 as a factor?

8. _______ What is the greatest whole number less than 150 that has an odd number of distinctpositive factors?

9. _______ Find n such that 2! � 3! � 4! � n = 8! .

10. ______ What is the smallest positive integer n for which 72 is a factor of n! ?

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Geometry Stretch

1. 25 (G, P)

2. 12 (F, M)

3. (F)

4. 118 (G, E, F)

5. (F)

6. 18 (T, S)

7. (F)

Answers8. 4 (M, P)

9. 238 (S, F)

10. 576 π (M, F)

Algebra Stretch

1. 6 (G, M, E)

2. 9702 (P)

3. 29 (C)

4. 20 (C, G)

5. 59 (C, P, G)

6. -24 (F, G)

7. -3 (M)

Answers8. 10 (F, G, E)

9. 94 (F, G, E)

10. 48 (P, F)

Factoring Stretch

1. 17 (C)

2. 16 (F, E, G)

3. 3 (P, G, E)

4. (P, E)

5. 36 (F, P)

6. 84 (C)

7. 555 (E)

Answers8. 144 (P, E)

9. 140 (C, E, P)

10. 6 (P, E)

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©MATHCOUNTS 2001�02 1

2001-2002 MATHCOUNTS School Handbook

WARM-UPS

The Warm-Ups contain problems that generally survey the middle school mathematics curriculum. Foruse in the classroom, the problems in the Warm-Ups serve as excellent additional practice for themathematics that students are already learning. In preparation for competition, the Warm-Ups can be usedto prepare students for problems that they will encounter in the Sprint Round.

Answers to the Warm-Ups include one-letter codes, in parentheses, indicating appropriate problem-solving strategies. However, students should be encouraged to find alternative methods of solving theproblems; their methods may be better than the one provided! The following strategies are used:C (Compute), F (Formula), M (Model/Diagram), T (Table/Chart/List), G (Guess & Check), S (SimplerCase), E (Eliminate) and P (Patterns).

MATHCOUNTS Symbols and Notation

Standard abbreviations have been used for units of measure. Complete words or symbols are alsoacceptable. Square units or cube units may be expressed as units2 or units3.

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WARM-UP 1(For #1 - #4) Glacier Park Lodge was built in 1912 in Montana. Sixty columns support the verandasand form a colonnade in the lobby. Each column is made from a gigantic fir or cedar tree 500 to800 years old that still retains its bark. Mule teams dragged tree trunks from the railhead to thebuilding site. Each has a diameter of 36 to 42 inches, a height of 40 feet and a weight of 15 tons.

1. ________ Each flat car on a train held two of these columns. What was the total weight, inpounds, of the two columns loaded on a flat car?

2. ________ What is the minimum circumference of a column rounded to the nearest inch?

3. ________ What is the maximum circumference of a column rounded to the nearest inch?

4. ________ A representative column has diameter 42 inches, height 40 feet and weight 15 tons.What is the mean number of pounds per cubic foot? Round your answer to thenearest whole number.

5. ________ What is the product of all the even integers from �6 to 7,inclusive?

6. ________ Rick has 6 different pairs of socks. What�s the probability thattwo randomly selected socks will be from a matching pair?Express your answer as a common fraction.

7. ________ For what single digit value of n is the number n5,3nn,672 divisible by 11?

8. ________ Suppose that a * b = a + b + ab. If x * 1 = 5, what is the value of x?

9. ________ Mrs. Smale�s class filled out a survey. Here are the results for her 30 students.

14 like hot dogs18 like cheeseburgers16 like tacos8 like both hot dogs and cheeseburgers7 like tacos and cheeseburgers6 like tacos and hot dogs1 likes none of these

How many students like all three?

10. _______ What is the sum of the positive integer factors of 225?

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WARM-UP 1

1. 60,000 (C)

2. 113 (F)

3. 132 (F)

4. 78 (F)

5. 0 (C, P,T)

6. (F, T)

7. 3 (E, P)

8. 2 (C, F)

9. 2 (M, G,F)

10. 403 (T, P,F)

Solution � Problem #6Imagine Rick picking the socks one at the time. For his first pick, he can choose any sock, and

still have an equal chance of getting a match on his second pick. So his first pick really doesn�tenter into the answer. For his second pick, he has 11 socks left to choose from, only one of whichwill make a match. So the probability of picking the other sock that will make the match is 1/11.

Connection to... Tests for divisibility (Problem #7)You probably already know a few �tricks� or divisibility rules for numbers like 2, 3, 4 and 5. To

see if a number is divisible by 3, you just have to add all of the digits of the number together andsee if the sum of the digits is divisible by 3. There is a similar test for divisibility by 11. We cancalculate a number�s alternating digit sum and see if that sum is divisible by 11. For example, to seeif the number 25,949 is divisible by 11, we can alternate putting subtraction and addition signsbetween the digits, always starting by putting a subtraction sign after the first digit: 2-5+9-4+9.This alternating digit sum comes out to 11, which is divisible by 11, so 25,949 is divisible by 11. For#7, we would get n - 5 + 3 - n + n - 6 + 7 - 2. By simplifying the expression to n - 3, you can find thevalue(s) for n which would make the alternating digit sum divisible by 11. Remember zero is divisibleby 11. How many other numbers do you know divisibility tests for?

Investigation & Exploration (Problem #10)To understand the factors of 225, consider its prime factorization: 52�32. All of the factors of

225 can now be viewed in a table by making row and column headings out of the breakdown of thetwo unique prime factor parts (52 and 32). To fill in the table, multiply the corresponding row andcolumn headers together. The interior of the table will show all of the factors of 225:

1 3 32

1 1 3 9 5 5 15 45 52 25 75 225

Using this method, try to find all of the factors of 72. How is finding the number of factors in4500 the same and different? How many different, positive factors does 4500 have? Can you comeup with a method for determining how many different, positive factors a number has if you are givenits prime factorization?

Answers

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©MATHCOUNTS 2001�02 4

1. ________ The skull of a Tyrannosaurus Rex found in 1990 weighed a ton and was 5 feet long.What was the mean number of pounds per linear foot?

2. ________ For a recent year, the average wind speed, in miles per hour, for the ten windiestU.S. cities are listed below:

Amarillo, Texas 13.5 Blue Hill, Massachusetts 15.4Boston, Massachusetts 12.5 Casper, Wyoming 12.9Cheyenne, Wyoming 12.9 Dodge City, Kansas 14.0Goodland, Kansas 12.6 Great Falls, Montana 12.7Lubbock, Texas 12.4 Rochester, Minnesota 13.1

What is the median of the average wind speeds, in miles per hour, for these tencities? Express your answer as a decimal to the nearest tenth.

3. ________ In the hexagonal lattice shown to the right, each point is one unit fromits nearest neighbor. How many equilateral triangles can be drawnusing a combination of three of the lattice points as vertices?

4. ________ The number 1 is both a smute and thripe. If the integer s is a smute, then the nextsmute is s + 5. If the integer t is a thripe, then the next thripe is 2t + 1. What isthe smallest whole number greater than 1, that is both a smute and a thripe?

5. ________ In 1967, about 900 eagles were believed to exist in the continental 48 states. Morethan 200,000 are now believed to be present. What is the smallest whole numberfactor that the number of eagles in 1967 could have been multiplied by to yield morethan 200,000 eagles now?

6. ________ The sum of the squares of two consecutive positive integers is 85. What is the sumof the two integers?

7. ________ Dave and Nick share their bread with Albert. Dave has 5 loaves of bread and Nickhas 3 loaves. They share the bread equally among the three of them. Albert givesDave and Nick $8, which they agree to share in proportion to the amount of breadthey each gave away. How many dollars should Dave receive?

8. ________ Let x be a positive number and y be its reciprocal. Compute .

9. ________ A rhombus is formed by two chords and two radii of a circle withradius R meters. What is the number of square meters in thearea of the rhombus? Express your answer as a commonfraction in terms of R.

10. _______ In a bag, there are 3 red marbles and B blue marbles. Two marbles are randomlyselected from the bag without replacement. The probability that the two marblesare the same color is 0.5. Calculate the sum of all possible values of B.

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©MATHCOUNTS 2001�02 5

WARM-UP 2

1. 400 (C)

2. 12.9 (C)

3. 8 (P)

4. 31 (P, T)

5. 223 (C)

6. 13 (G, F, E)

7. 7 (T, E)

Solution � Problem #8This problem can be handled with a basic knowledge of algebra and fractions. Notice the

progression of the expression if we just substitute 1/x for y and carry out all of the addition:

Seems like there must be an easier way...and there is! Make a simpler case out of the problem.The problem has to work for any x and y that fit the requirements. Therefore, let�s pick x = 2 andy = 1/2. Plugging these values into the expression will lead to a much simpler solvingprocess...especially if you are able to use a calculator!

Solution � Problem #10

3 Red marbles B Blue marbles This demonstrates three ways to pick different

colored marbles if B=1.

Each N represents the number of equally likely occurences of the events.

Ns(same color) = Nd(different colors) Since the probability of pulling twothe same color is .5.

Nr(both Red) + Nb(both Blue) = Nd(different colors) 3 +

BC

2 = 3·B For every blue marble, there would be 3 red marbles

that could be paired with it. See picture above. 3 + = 3B

6 + B(B �1) = 6B

B2 � 7B + 6 = 0 and then by trial and error, or by factoring, B=1 or B=6.

Sum = 1 + 6 = 7.

Investigation & Exploration (Problem #4)Make a list of the first seven smutes. Do you see a pattern? Can you find a formula, in terms

of x, so that you could find the xth smute number? Make a list of the first seven thripes. What is apattern that you see in this list of numbers? Again, can you develop a formula, in terms of x, so thatyou could find the xth thripe number? Will any other numbers be both smutes and thripes?

Answers

8. 1 (C, S)

9. (R2 � ) / 2 (F)

10. 7 (P, G)

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©MATHCOUNTS 2001�02 6

WARM-UP 31. ________ The stem-and-leaf plot below reports the number of months spent on 20 separate

investigations spanning the 1980�s and 1990�s. Find the positive difference betweenthe median number of months spent on investigations during the 1980�s and duringthe 1990�s. Express your answer as a decimal to the nearest tenth.

Number of Months Spent on Separate Investigations 1980�s 1990�s

9 8 7 0 7 5 4 3 2 1 1 0 1 0 5 5 9

2 1 3 0 3

4 8 9

2. ________ The slope of the line tangent to the graph of y = x2 at the point (3,9) is 6. What isthe y-intercept of this tangent line?

3. ________ A natural gas pipeline ruptured, triggering an explosion. The amount of soil requiredto fill the hole made by the explosion was equivalent to the amount of soil in arectangular prism 86 feet long, 46 feet wide and 21 feet deep. How many dumptruck loads, of 20 cubic yards of soil each, were needed to transport the soil?

4. ________ Alex has four bags of candy, weighing 1, 2, 3 and 4 pounds. To arrange them, he willpick up two bags which are next to one another, compare their weights on a balancescale and put them back, with the heavier one to the left. What is the maximumnumber of times he could possibly have to swap two bags before he has the bags inorder from heaviest to lightest?

5. ________ The cube of the three-digit natural number A7B is 108,531,333. What is A+B?

6. ________ A fraction is equivalent to 3/5. Its denominator is 60 more than its numerator.What is the numerator of this fraction?

7. ________ Each small square in the figure to the right has an area of6 cm2. What is the number of square centimeters in thearea of the shaded region?

8. ________ �A plumber and his helper leave the shop at 8:20 A.M. to repair a faucet. Theyreturn at 11:10 A.M. They charge 60 cents an hour for the time the plumber is gonefrom the shop, half as much for his helper�s time, and 85 cents for all of thematerials� (paraphrased from The Thorndike�s Arithmetics � Book Three, C.Thorndike, 1917). What was the number of dollars charged? Express your answer asa decimal to the nearest hundredth.

9. ________ What is the number of units in the length of segment PQ? Express your answer as amixed number.

10. _______ What is the smallest positive integer that has 8, 30 and 54 as factors?

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WARM-UP 3

1. 8.5 (C)

2. -9 (F, M)

3. 154 (F)

4. 6 (M, P)

5. 11 (E, S)

6. 90 (P, G, C)

7. 36 (F)

Solution � Problem #7Sometimes the easiest way to find the answer to a problem is to find the answer to a different,

but related, problem first. In this case, it may be easier to find the areas of the non-shadedregions and subtract that from the entire region. These non-shaded regions will help us to answerthe question about the shaded region.

The area of the entire figure is 3 x 5 x 6 = 90 cm2. (Remember there are just 15 squareregions, but each of those has an area of 6 cm2.)

The area of the three non-shaded regions are:I. the right triangle at the top: 0.5 x 5 x 2 x 6 = 30 cm2.II. the right triangle at the bottom, on the right:

0.5 x 1 x 3 x 6 = 9 cm2.III. the region at the bottom, on the left (one square + one half-square + one right triangle): 6 + 3 + 0.5 x 1 x 2 x 6 = 15 cm2.Therefore, the area of the shaded region is 90 � 30 � 9 � 15 = 36 cm2.

Connection to... Statistics (Problem #1)In statistics there are a few different values we can use that are all considered measures of

central tendency. The measure of central tendency that we are using in this problem is the median,or the middle number once the data has been ordered. Probably the most common measure ofcentral tendency is the mean or the average. Many students are very familiar with how to find theaverage of a data set. What about the mode of a data set? How would you explain to someone howto find the mode? In a perfect bell curve, the median, mean and mode are all the same value. Canyou create a data set where these three measures of central tendency are all the same?

Investigation & Exploration (Problem #4)Programmers often need to sort a list of objects, whether numbers, names or some other set of

information. A bubble sort, the type of sort described in this problem, systematically goes throughthe data comparing only two adjacent values, swapping the values if necessary and then starting thecheck again at the beginning until no more swaps can be done. It is called a bubble sort because thelarger values are supposed to slowly �bubble up� toward the top of the list. This kind of sort is verytime-consuming when there are a lot of data to sort. Can you create a formula that will tell you howmany swaps will be needed in a worst-case scenario for n objects, meaning n objects are currentlysorted in the exact opposite order of how you want them arranged?

Computer scientists have discovered other methods of sorting that are much more efficient.Explore what some of these other methods of sorting are and how they work.

Answers

8. 3.40 (C, T)

9. (C)

10. 1080 (P, T, C)

� ��

I

IIIII

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©MATHCOUNTS 2001�02 8

WARM-UP 41. ________ Given that 4a+5b+7c = 13 and 4a+3b+c = 19, what is the value of a+b+c?

2. ________ During an 8-month Dungeoness crab season, 325 crabbers in Newport, Oregon landed4,913,977 pounds of crab and sold them for $2.00 per pound. What wasthe average number of dollars earned by each crabber per monthduring the 8-month crab season? Express your answer to the nearestdollar.

3. ________ Jane invests $5000. She expects to have $5000(1.06)8 after eight years. Herinterest rate, compounded annually, is n%. What is the value of n ?

4. ________ How many factors of 1800 are multiples of 10?

5. ________ Two cards are randomly selected without replacement from a set of four cardsnumbered 2, 3, 4 and 5. What is the probability that the sum of the numbers on thetwo cards selected is 7? Express your answer as a common fraction.

6. ________ Point P is on the number line. The distance between zero and P is four times thedistance between P and 30. What is the sum of the two possible values for P?

7. ________ How many digits are printed by a printer that prints all the whole numbers from1 to 728, inclusive?

8. ________ What is the number of inches in the sum of the perimeters for the two similartriangles shown?

9. ________ The vertices of a triangle are (0,0), (0,y) and (x,0), where both x and y are greaterthan zero. The area of the triangle is 30 square units and the perimeter is also30 units. What is the value of x + y?

10. _______ Yesterday, 2000 circular jellyfish, each of diameter 2 feet, were lying within asection of ocean floor that measured 120' by 360'. None of the jellyfish wereoverlapping each other. What percent of that section of ocean floor did thejellyfish cover? Express your answer to the nearest whole number.

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WARM-UP 4

5. 1/3 (F, T)

6. 64 (M, G)

7. 2076 (T, P)

Solution � Problem #4Solution 1: Each factor that is a multiple of 10 must have at least one factor of 2 and one

factor of 5. There are three factors with at least one factor of 2 (2,4,8). There are two factorswith at least one factor of 5 (5, 10). There are three possibilities for a power of 3 (1,3,9).Therefore, there are 3 x 2 x 3 = 18 factors of 1800 that are multiples of 10.

Solution 2: The prime factorization of 1800 is given by 1800 = 23�32�52. Therefore, thefactors of 1800 all have factorizations of the form 2i�3j�5k, where .In order to be a multiple of 10, both i and k must be at least 1. Therefore, there are 3 x 3 x 2 = 18factors of 1800 that are multiples of 10.

Connection to... Finding the area of a triangle (Problem #9)

There are several different methods for finding the area of a triangle. A = �� bh, where

b = length of the base and h = the height of the triangle is one formula. Additionally, A = �� ab(sin C)

where a,b = length of two consecutive sides and C = measure of the angle included between the sidesis another formula.

The formula for the area of an equilateral triangle where s = the length of one side is A = .

Heron�s Formula: A = where s = the semi-perimeter (half the perimeter) and

a, b and c are the length of the sides, is used when only the lengths of the sides are known.Research how each of these formulas for the area of triangles was derived.

Investigation & Exploration (Problem #2)Consider the following:a. If the crabber is unemployed the remaining months of the year, what would the mean

monthly income be if figured on a 12-month basis?b. How might you find statistics on income for fishermen to determine how typical this is?c. How many hours per day did a crabber likely work during the 8-month season?d. How many days per week did a crabber likely work during the 8-month season?e. Using your answers from c and d, estimate the hourly wage for a crabber.

Answers8. 60 (F, P)

9. 17 (M, F, G)

10. 15 (F)

1. 4 (G, P)

2. 3780 (F)

3. 6 (F)

4. 18 (P, E)

V� ��

V V D V E V F− − −1 62 71 6

� � � � � �≤ ≤ ≤ ≤ ≤ ≤L M N� �

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WARM-UP 51. ________ The units digit of n3 is 3, and n is an integer. What is the

units digit of n?

2. ________ What is the number of inches in the perimeter ofquadrilateral ABCD?

3. ________ A square has a side length of x units. The square�s length is then increased by 2 unitsand its width is increased by 9 units. By how many square units does the area of thenew rectangle exceed the area of the square? Express your answer in terms of x.

4. ________ The diagram shows six congruent circles with collinear centers on the x-axis. Howmany paths of length 3 are there from A=(0,0) to B=(6,0) if the paths must remainon the circumferences of the circles?

5. ________ The Los Angeles Unified School District predicts that enrollment in its schools willincrease from 711,000 in 2000 to 750,000 in 2005. If one teacher is needed foreach increase of 30 students, how many more teachers will be needed?

6. ________ Aimee tosses one fair 6-sided die labeled 1 through 6 and one fair 4-sided dielabeled 1 through 4. What is the probability that the sum Aimee rolls is less than5? Express your answer as common fraction.

7. ________ An arithmetic series is called a concatenation series if the sum of the series isrepresented by the concatenation of the first and last terms. For example17+19+21+� +85 = 1785. Find a concatenation series with 41 consecutive integers anda four-digit sum. What is the sum of the integers of this series?

8. ________ What is the largest possible value of a + b + c + d in the prime factorization treeshown?

9. ________ �A laborer was hired for a year, to be paid $80 and a suit of clothes. After heworked 7 months, he left. Therefore, he only earned 7/12 of his yearly salary. Forhis wages, he received the suit of clothes and $35. What was the dollar value ofthe suit of clothes?� (paraphrased from an 1848 text by Joseph Ray, Ray�s Algebra)

10. _______ Solve 22 � 42 � 82 � 162 � � � � 10242 = 2x for x.

π

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WARM-UP 5

5. 1300 (C)

6. 1/4 (T, P)

7. 1353 (G, E, F)

Solution � Problem #3A drawing may help you visualize the problem and solution.The drawing can illustrateü the original square with an area of x2 square unitsü a rectangle below the original square with a width of 2

and a length of x for an area of 2x square units,ü a rectangle on the side of the original square with a

width of x and a length of 9 for an area of 9x squareunits

ü a rectangle that connects the other two additions witha width of 2 and a length of 9 for an area of 18 squareunits.

When these areas are combined we get x2 + 2x + 9x + 18 or x2+11 x + 18. The area could also befound by finding the product of the length and width, (x + 2)(x + 9) = x2+ 11x + 18. Therefore, thenew rectangle increased in area by 11x + 18 units.**Note that this illustrates an area model for the product of two binomials.

Connection to... History (Problem #9)Problems from old mathematics textbooks provide glimpses into the social and economic history

of the United States. What was the laborer�s annual wage? Weekly wage? How long would it takeyour students to earn $108? Does your students� pay ever include goods or food?

One way to compare the cost of goods over time is to compare �the number of hours of workneeded to purchase a particular item�. In 1848, how many weeks did it take the laborer to �earn�the suit of clothes? Ask your students to estimate the cost of a �suit of clothes� now and ask themhow many hours or weeks it would take for them (or their parents) to �earn� it. Check outhttp://www.westegg.com/inflation/ for a calculator that will show how inflation has affected thevalue of a dollar over time.

Investigation & Exploration (Problem #5)Demographics is the study of population characteristics. The results of such studies are used

to make decisions regarding your community. Is the population of your community increasing,decreasing or remaining the same? Is the number of students in your school district increasing,decreasing or remaining the same? What might be contributing factors to the population growth ordecline of your community?

Answers8. 43 (P, E)

9. 28 (F, G)

10. 110 (P, F)

1. 7 (P, E, T)

2. 96 (F)

3. 11x + 18 (T, M)

4. 64 (P, C, S)

x2 9x

2x 18

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©MATHCOUNTS 2001�02 12

WARM-UP 61. ________ The diameter of a large clock hanging in a mall in Melbourne, Australia

is 3.1 meters. What is the circumference of the clock in meters?Express your answer as a decimal to the nearest tenth.

2. ________ On a particular day, 100 airplanes depart from a Babbage airport. Ten of the planesare delayed by an hour each. Of the remaining planes, half are on time and half aredelayed by 20 minutes. What is the number of minutes in the average flight delay?

3. ________ Tickets numbered 1 through 100 are placed in a bag and one is randomly drawn.What is the probability that the factors of the number drawn will include 2, 3 and5? Express your answer as a common fraction.

4. ________ What is the units digit of the sum: (1!)2 + (2!)2 + (3!)2 + (4!)2 + ... + (10!)2 ?

5. ________ How many integers can be written as the sum of three different members of the set{2, 4, 6, 8, 10, 12, 13} ?

6. ________ A painted 2x2x2 cube is cut into 8 unit cubes. What fraction of the total surfacearea of the 8 small cubes is painted?

7. ________ Otto starts out facing due north. He turns to the right, first by ten degrees, thenby twenty degrees, then by 30 degrees, increasing his turn by 10 degrees each time.He continues this process until he is again facing due north. How many degrees doesOtto rotate in the last turn before he stops?

8. ________ Lines L, M, N, P, Q and R are drawn on the xy-plane. Theslopes of the six lines (not in order) are: 2/3; 2; 3; -3/2;-1/3; -1/2. Which of the six lines (L, M, N, P, Q or R) hasa slope of -1/2 ?

9. ________ The point (a, b) is reflected over the x-axis. The coordinates of the new point are(c, d). What is the value of ab + cd ?

10. _______ In the hexagonal lattice shown, each point is one unit from itsnearest neighbors. In the same plane, how many circles of radiusone unit pass through at least two points of the lattice?

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WARM-UP 6

5. 19 (P)

6. 1/2 (M, F, P)

7. 80 (T, G)

Solution � Problem #10For this kind of counting problem, it is best to approach it with a plan of how we are going to

count the objects so that we are sure to not miss any! First, let�s find all of the circles of radiusone unit that would have their center outside of the lattice points. These sixcircles are the only possibilities. Any circle with a center further out wouldeither not reach any of the lattice points or would not pass through two of thepoints. Any circle closer in would also not pass through two lattice points.(*Notice the centers of these circles are on the perpendicular bisectors of thesegments formed by each pair of consecutive outer lattice points.)

Now let�s count the number of circles that have their center on the outerlattice points. Since the lattice points are one unit from each of theirclosest neighbors, then the circle would pass through the three closestneighbors of whichever outer point we picked as the center. There are six ofthese circles.

And finally, if the center of the circle was inside the outer lattice points, it couldonly be at the middle lattice point. The circle now goes through each of the other sixlattice points. From this picture, we can see that if we shifted the center of the circle

to any other interior location, the circle would not pass through two lattice points.

There are 13 circles of radius one unit, in the same plane, that pass through at least two pointsof the lattice.

Connection to... Geometric Representations (Problem #8)Slopes provide a visual, geometric method for ordering fractions. To compare 2/3 and 4/5, lines

may be drawn with these slopes. Looking at the steepness of the lines helps to put the fractions inorder. However, what do you notice about the steepness of a line with a slope of -3/4 compared tothe steepness of a line with a slope of 3/4? Graph two lines with these slopes. If you were a skier,which line would you rather ski down? Which one is steeper? How can you tell if a line has anegative or positive slope when looking at a graph?

Investigation & Exploration (Problem #6)Answer the same question for a 3 by 3 by 3 cube. Drawing a picture may help, but can you then

figure out a formula for answering the problem if we change it to a 4 by 4 by 4 cube? What happenswhen you use an n by n by n cube? Does the ratio of the unit cubes� painted surface area to totalsurface area increase, decrease or remain the same? Could you write a paragraph to a classmateexplaining why this happens?

Answers8. Q (M, E)

9. 0 (G, M)

10. 13 (P)

1. 9.7 (F)

2. 15 (F, T)

3. (E)

4. 7 (P, C)

����

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WARM-UP 71. ________ During the 1995 season, Oregon crabbers had a 15 million pound Dungeoness Crab

harvest which sold for $24.7 million. That record was broken during the 2000season when 15,616,728 pounds of the crabs were harvested and sold for$31,297,583. How many more cents per pound did crabs sell for in the 2000 seasonthan the 1995 season? Express your answer to the nearest whole number.

2. ________ How many of the first 80 positive integers can be written as the sum of two distinctpowers of 3? For example, 28=27+1=33+30 can be so expressed, but 29 cannot be.

3. ________ A 3 inch by 4 inch rectangle is rotated about a corner. What is the maximumnumber of square inches in the area of the region touched by some point of therectangle as it makes a full rotation? Express your answer in terms of .

4. ________ The length, width and height of a rectangular box are each decreased by 50%. Bywhat percent, to the nearest tenth, is the volume of the box decreased?

5. ________ A raffle was held and 1200 tickets were sold for $2.50 each. There were 17 winners.The first-prize winner received $1000. The four second-prize winners each received$250. The remaining winners each received $50. What percent of the total ticketsales was profit? Express your answer to the nearest whole number.

6. ________ Suppose that Keith�s average score on four English tests is 85. The average of histhree highest scores is 88.5 and the average of his three lowest scores is 82.5.What is the average of his highest and lowest test scores? Express your answer as adecimal to the nearest tenth.

7. ________ Let a*b = the least common multiple of a and b. What is the sum of all naturalnumber values of x such that 15*x = 45 ?

8. ________ Let p and q be different prime numbers. How many positive factors will (p2q4)3

have?

9. ________ On a trick 6-sided die the probability of rolling a 1 or 2 is each 1/4, the probabilityof rolling a 3 or 4 is each 1/6 and the probability of rolling a 5 or 6is each 1/12. The trick die and a standard die are rolled. What isthe probability of rolling a sum of 7? Express your answer as acommon fraction.

10. _______ Choose a number. Triple the number. Add 200. Double the result. Subtract 100.Divide by 4. Subtract 150% of the original number. What is the value of the result?

π

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WARM-UP 7

5. 13 (C)

6. 83.5 (P, T, C)

7. 54 (E, T)

Solution � Problem #9One way to solve this problem is to consider each situation on the trick die combined with the

needed value on the regular die. Trick die Regular die

1 6 (1/4 x 1/6) = 1/242 5 (1/4 x 1/6) = 1/243 4 (1/6 x 1/6) = 1/364 3 (1/6 x 1/6) = 1/365 2 (1/12 x 1/6) = 1/726 1 (1/12 x 1/6) = 1/72

Total 1/6

A more efficient solution is to recognize that no matter what value is rolled on the trick diethere is a 1/6 chance that the correct value to make a sum of 7 will be rolled on the regular die.

Solution � Problem #101. Call the original number N.2. Tripling gives us 3N.3. Adding 200 yields 3N + 200.4. Doubling produces 6N + 400.5. Subtracting 100 leaves 6N + 300.

6. Dividing by 4 yields (3/2) N + 75.

7. Subtracting 150% of the original number N gives us (3/2)N + 75 -150%N=(3/2)N+75 -(3/2)N =75

Therefore, the answer is 75. Notice how the answer is not dependent on your original value ofN. This type of problem is often a part of �trick� mathematical problems. Can you design a similartype of problem where the final answer is 2002?

Investigation & Exploration (Problem #2)What if we change the question to ask how many of the first 250 integers can be written as a

sum of two distinct powers of 3? Can you find a pattern or formula that may help you to find theanswer if we change the question to cover the first 8000 positive integers? Change the question toask how many of the first 8000 positive integers can be written as a sum of three distinct powersof 3? How does your knowledge of combinations help you?

Answers8. 91 (T, P, F)

9. 1/6 (T)

10. 75 (F, G)

1. 36 (C)

2. 6 (P, E, T)

3. 25 (F, M)

4. 87.5 (S, F)

π

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WARM-UP 8(For #1 and #2) The following data is reported by a company, Wheat Montana Farms: one bushel ofwheat contains 1 million individual kernels which yield 42 pounds of white flour, from which 73loaves of bread are made. One loaf of bread yields 16 slices from which 8 sandwichesare made.

1. ________ How many sandwiches can be made from one bushel of wheat?

2. ________ A Wheat Montana combine can harvest 1000 bushels of wheat per hour,on average. The amount harvested per minute, on average, could producehow many loaves of bread? Express your answer to the nearest hundred.

3. ________ Ray reads 18 pages the first day, 23 pages the second day, 28 pages the third day,33 pages the fourth day and continues to add 5 more pages each successive day.How many pages will Ray read in the first fourteen days of his reading program?

4. ________ A bag contains numbered tags 1, 2, 3, 4, . . . , 200. One tag is selected at random.What is the probability that the number on the tag selected is a multiple of 3 or 7?Express your answer as a common fraction.

5. ________ Two sides of an isosceles triangle have measures of x + 10 and x + 40 and theperimeter of the triangle is 420 units. Find the sum of the two possible values of x.

6. ________ What is the units digit of 610�512 ?

7. ________ What value of x, in pounds, will make the lever be in balance at the fulcrum?

8. ________ Three adults noticed a large tree along The Trail of Cedars in Glacier National Park .Together they were able to stretch their arms to form a ring around the tree withtheir finger tips just meeting. Their arm spans were 6 feet 1 inch, 5 feet 8 inchesand 5 feet 5 inches. What is the number of inches in the diameter of this cedar?Express your answer to the nearest whole number.

9. ________ A sandbox in the shape of an equilateral triangle is 10 meters on each side. A fenceis built around the triangular sandbox at a constant distance of 4 meters. What isthe number of square meters in the area between the fence and the triangularsandbox? Express your answer in terms of π .

10. _______ A regular hexagon is inscribed in a unit circle. What is the number of square unitsin the area of the hexagon? Express your answer as a fraction in simplest radicalform.

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WARM-UP 8

5. 230 (F)

6. 0 (P, S)

7. 10 (F)

Solution � Problem #9

The figure to the right shows our triangular sandbox and the fence aroundit. Notice that the area between the fence and the sandbox is divided into sixpieces. The corners are portions of circles while the sides arequadrilaterals. The fence will be linear along the sides of the triangle,running parallel to them. The corners of the fence will be curved, since allpoints 4 meters from a vertex of the triangle will form a circle.

Now we need to determine how large the circular and quadrilateralpieces are. Notice the quadrilaterals must be rectangles. Look at thepiece with the label of 4m. It must be going perpendicular, otherwisethe distance would be greater than 4m. So each rectangular regionmeasures 4m x 10m or 40m2. Since there are three of these rectangular regions, they total 120m2.

Now for the circular regions. How much of a whole circle is each section? We can figure thisout by finding the central angle. Let�s look at the region at the top. There are four angles with thepoint at the top of the triangle as their vertex point. Two of those angles are right angles (fromthe rectangles) and one of the angles is 60° (from the equilateral triangle). That leaves the anglefor the circular region to be 120°. That is 1/3 of a circle. Since there are three of these regions,together they will form a full circle, whose radius is 4m. So the total area for the circular regionsis 16π square meters.

Putting together the rectangular and circular regions, we end up with an area equalto 120 + 16 square meters.

Connection to... Weeding out unnecessary information (Problem #1)In the introductory paragraph for problem #1, there is a lot of information given that is really

not necessary. Sometimes the most difficult part of a problem is picking out the information that istruly necessary in calculating the answer. Notice that the number of kernels, pounds of flour andslices of bread are all given, but only distract you from the information you need to answer #1. Allyou need is that a bushel of wheat will make 73 loaves of bread and one loaf of bread will make 8sandwiches. Whether answering comprehension questions after reading an essay or answering wordproblems in math class, be sure to weed out any information that is not useful, so that you are notdistracted by it. Distracting the audience with unnecessary details or actions is the idea behindmany riddles, tricky test questions and even magic tricks...it works on a lot of people!

Investigation & Exploration (Problem #8)Measure the height and arm spans of the students in your class. Make a table and a scatter plot

to compare the height and arm spans (let Height be the x-axis and Arm Span be the y-axis). Fit aline to your data. How well does your data match your line? How do the heights and arm spanscompare in the group of people chosen? Can you come up with an equation for your line of best fit?Enter your data into a graphing calculator and see how close your equation is to your calculator�sequation for the line of best fit.

Answers

8. 66 (F)

9. (F, M)

10. (F, M)

1. 584 (C)

2. 1200 (C)

3. 707 (F, P)

4. (P, E)

��� ��+ π

� ����

��

π

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©MATHCOUNTS 2001�02 18

WARM-UP 91. ________ A bag contains 30 red marbles and 50 white marbles. Twenty percent of the red

marbles are removed, and each removed red marble is replaced by 4 white marbles.Next, fifty percent of the white marbles are removed, and each removed whitemarble is replaced by 2 red marbles. In the bag, what is the ratio of white marblesto red marbles after replacements? Express your answer as a common fraction.

2. ________ The table below shows the 1992 and 2000 salaries for employees with 10 years ofexperience at Seth�s Surf shop. What is the positive difference between thepercentage of increase in pay for a Level A employee and a Level B employee from1992 to 2000? Express your answer to the nearest whole number.

1992 2000 Level A employee with 10 years of experience $36,090 $46,202 Level B employee with 10 years of experience $17,161 $22,338

3. ________ The five-digit whole number 3a,7b1 is a perfect square. What is the greatestpossible value for the product ab ?

4. ________ Adele and Toby are planning to build a new barn for theirhorses and want to cover the roof with tile toprotect it from fire. The roof will have a slope of5/12 known as a 5-12 pitch. Using the estimate thateach tile covers a region measuring 8 x 12 inches,what is the number of tiles needed to cover just theshaded portion of the roof shown?

5. ________ A rectangle has integer side lengths and its area is equal to 24 square units. Thelength of each side of the rectangle is increased by one unit. What is the largestpossible number of square units in the area of the new rectangle?

6. ________ The perimeter of a square is 60% of the perimeter of a triangle whose sidesmeasure 43, 47 and 50. What is the number of square units in the area of thesquare?

7. ________ What is the value of the expression when n = 12?

Express your answer as a common fraction.

8. ________ Zan tosses one fair 6-sided die with faces labeled 1 through 6 and one fair 4-sideddie with faces labeled 1 through 4. If at least one die shows a �3�, what is theprobability that the sum is 5? Express your answer as a common fraction.

9. ________ Jen has 15 United States coins worth 76 cents. Using any combination of pennies,nickels, dimes, quarters or half-dollars, how many different, possible combinations ofcoins could Jen have?

10. _______ While driving, Ron averages r miles per hour for d miles and then averages 2r milesper hour for d miles. Overall, Ron�s average speed equals kr miles per hour. What isthe value of k ? Express your answer as a common fraction.

� ��

� ��

� ��

� ��

−������ −���

��� −���

��� −

+���

���...

Q

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©MATHCOUNTS 2001�02 19

WARM-UP 9

5. 50 (P, M)

6. 441 (F, M)

7. 1/13 (P, C)

Solution � Problem #10There are at least two good approaches to this problem that we recommended in the answer key

above. The first method for solving the problem is to use a formula.

The equation that will get us to our answer is Average speed =

In order to find the total time, we need to add the times it took for each part of the trip.Remember that time can be found by dividing the distance by the rate of speed. So by plugging inthe values from the problem we find that...

Average speed = = . Therefore k = 4/3

The next method of solving uses the idea of creating a simpler case. Since the problem mustwork for any choice of d and r, select �nice� numbers. For example, let�s let d = 40 miles and r = 20miles per hour. We still need to know how an average speed is found, but we won�t need to work withall of the variables.

Average speed = =

Therefore, .

Connection to... Interpreting Data (Problem #2)People often use data when negotiating contracts. A Level B employee might use the data to

argue that the pay for Level B employees had increased slightly more than $5,000 in 8 years whilethe pay for Level A employees had increased more than twice that, or more than $10,000. ALevel A employee might argue that the increase in Level B employees� pay had been 30 percent whilethe increase in Level A employees� pay had been 28 percent over the last 8 years. It has been saidthat you can make data say whatever you want it to say. Look in some of your recent, localnewspapers for examples of charts or graphs that may be misrepresenting data.

Investigation & Exploration (Problem #4)The building industry uses a lot of mathematics while planning construction projects. The slope

or pitch of a roof is the tangent ratio. The tangent is the ratio of the vertical change to thehorizontal change in a right triangle. Builders also need to consider the slope or pitches of otherthings they construct. Can you think of some other structures that have a slope? Considerpreparing land for a driveway to a house. In your neighborhood, take some measurements of someof the driveways and determine their slope. What about staircases or wheelchair ramps?Determine the pitch of these in your school. Are they all the same? What conditions might beconsidered in determining the pitch of a staircase?

Answers

8. 2/9 (T, P)

9. 5 (T, P)

10. 4/3 (S, F)

1. (P, M)

2. 2 (C)

3. 12 (P, E, G)

4. 780 (F)

����

G GGU

GU

GGU

U+

+= =

���

��

Total distance Total time

���

���

�� ��

= = =NU N N� �

�� ��� �

���

++

=

Total distance Total time

Total distance Total time

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©MATHCOUNTS 2001�02 20

WARM-UP 101. ________ A gallon of milk weighs 8.6 pounds. It takes 10 pounds of milk to make 1 pound of

cheddar cheese. How many pounds of cheddar cheese can be made from 100 gallonsof milk?

2. ________ The length of two sides of a triangle are 4 units and 7 units. The length of the thirdside is a whole number of units. What is the maximum possible length of the thirdside?

3. ________ Solve for a in terms of b if two more than twice a is three less than the square ofthe number which is one less than b.

4. ________ The variables c, d, e and f represent distinct digits in this 3 ccorrectly worked multiplication problem. What is X d 5the value of c + d + e + f ? 1 9 5

1 e 61 f 5 5

5. ________ Becky has ten brown socks and ten black socks. If she randomly selects two socksfrom the drawer simultaneously, what is the probability that they are the samecolor? Express your answer as a common fraction.

6. ________ How many ordered pairs (a,b) of positive integers satisfy the equation 3a + 4b < 12?

7. ________ A regular hexagon ABCDEF satisfies AB=1 unit. How many square units are in thearea of quadrilateral ABCE? Express your answer in simplest radical form.

8. ________ The figure is formed by beginning with a square with side length 1 (labeled with a 1)and attaching a congruent square (2).Then, at each successive step, a square isattached to the longer side of therectangle, making a new rectangle. Whatis the number of square units in the areaof square 7?

9. ________ How many triangles are in the figure shown?

10. _______ Suppose that x is chosen from the set {1, 2, 4, 8} and that y is chosen from the set{3, 6, 12, 24}. How many different values could x/y be?

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©MATHCOUNTS 2001�02 21

WARM-UP 10

1. 86 (C)

2. 10 (F, M)

3. (C)

4. 25 (E, G)

5. 9/19 (M)

6. 3 (E, G, T)

7. (F, M)

8. 169 (P)

9. 25 (P)

10. 7 (E, P, T)

Solution � Problem #10One way to solve this problem is to think in terms of the factorizations of the integers in each

set. Notice that the first set is {20, 21, 22, 23}, and the second set is {3·20, 3·21, 3·22, 3·23}. So,every fraction will be of the form 2a/(3·2b) = 2a-b/3. The value of a-b is 3, 2, 1, 0, -1, -2, or -3, sothere are 7 possible values of the fraction x/y.

Connection to... the Golden Ratio (Problem #8)Notice how the side lengths of the successive squares form the Fibonacci sequence (1,1,2,3,5...).

The spiral formed by putting a quarter circle in each square closely approximates the spiral formedby many natural forms, like seashells. For eachnew rectangle, the ratio of the height to thewidth approaches the so-called �Golden Ratio�(approximately 1.618034) which is widely seenboth in art and nature. Rectangles such as theseare known as golden rectangles and were believedto be the most aesthetically pleasing of allrectangles by the Ancient Greeks. Because ofthis, the golden rectangle can be found in manyexamples of their architecture.

Investigation & Exploration (Problem #3)

If Max worked out this problem and came up with as the answer and Carol worked the

problem and ended up with as the answer, are Max and Carol wrong? Can you show how

these two answers might be shown to be equivalent to the answer provided? Would any of the

answers be �more correct� than the others? Discuss the advantages and disadvantages of each

form of the answer.

Answers

D E E= − −�� �

E− −� �

�2 7

E E�

��− −

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©MATHCOUNTS 2001�02 22

WARM-UP 111. ________ A square has an area of 144 cm2. How many centimeters is each diagonal? Express

your answer in simplest radical form.

2. ________ When buying shirts at Sport�s Shirts, there is a fixed set-up fee and a constant costper shirt. The price for 20 baseball shirts would be $390. An order for 80 baseballshirts would cost $1110. How many dollars would 140 baseball shirts cost at Sport�s?

3. ________ From an 1855 text, School Arithmetic, by Charles Davies: �If twenty grains makeone scruple; three scruples make one dram; eight drams make one ounce; and twelveounces make one pound; what part of an ounce is 3/10 of a scruple?� Express youranswer as a common fraction.

4. ________ The number N is a positive multiple of both 6 and 8. N is also a factor of 432. Howmany different integers can N be?

5. ________ Sara is painting the four walls of her room. Her room is 8 feet wide by 10 feet long,and all of the walls are the same height. One can of paint covered exactly half ofone of the smaller walls. How many more cans of paint, of the same size, will sheneed to paint the rest of the room?

6. ________ The quantities x and y vary inversely and x = 27 when y = 9. Determine the valueof x when y = 60. Express your answer as a mixed number.

7. ________ A news article reported the length, in months, of investigations of jet accidentsinvolving commercial flights of U.S. carriers originating in the United States from1964-1996. The data is provided in the stem-and-leaf plot. What is the positivedifference, in months, between the median and the mode? Express your answer as adecimal to the nearest tenth.

Length, in months, of Investigations

0 5 6 6 6 7 7 7 7 8 8 8 8 8 8 9 9 91 0 0 0 0 0 1 1 2 2 2 3 3 3 4 4 4 4 4 5 5 5 5 5 6 7 7 8 92 1 1 3 7 83 0 44 95 4

8. ________ A coin of diameter 2 cm is dropped randomly on the tabletopshown so that the entire coin lies on the tabletop. Eachsquare tile is 10 cm on a side. What is the probability thatthe coin lies completely within one of the squares? Expressyour answer as a common fraction.

9. ________ Use each of the digits 2,3,4,6,7 and 8 exactly once to construct two three-digitintegers M and N so that M-N is positive and is as small as possible. Compute M-N.

10. _______ In this diagram, each short line segment has length 1. Theshortest paths from A to B each have length 3. How manypaths of length 4 are there from A to B? Assume that youcan change direction only at the four vertices.

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©MATHCOUNTS 2001�02 23

WARM-UP 11

1. (F, M)

2. 1830 (F, G)

3. 1/80 (C, M)

4. 6 (P, G)

5. 8 (F, M)

6. (C, F)

7. 4.5 (C)

Solution � Problem #9If we are trying to make M-N as small as possible, then we are trying to make the two numbers

as close to each other as possible. We can arrange for M-N to be less than 100 by choosing thehundreds digit for M to be one more than the hundreds digit for N . There are several ways to dothis. You also want to keep the tens digit of the greater number as little as possible, while keepingthe tens digit of the lesser number as large as possible. Using the 2 and 8 for the tens digits stillleaves the 6 and 7 for the hundreds digit. You may also try some trial and error and see that thisalso reveals that M=723 and N=684 are as close to one another as possible. Thus the answer weseek is 723-684 = 39.

Connection to... Science (Problem #6)Direct variations and inverse variations are used to explain the relationship between the volume

of a gas and temperature or pressure. The volume of a gas increases as the temperature increaseswhile the volume of a gas decreases as the temperature decreases, and the quantities are directlyproportional. The letter k represents the constant of proportionality. This is often representedalgebraically as V = kT or V1/V2 = T1/T2.

On the other hand, the volume of a gas decreases as the pressure exerted on the gas increaseswhile the volume of a gas increases as the pressure decreases. This illustrates an inverse variationand is often represented as V = k/p or V1/V2 = P2/P1.

Check a chemistry book to determine the names of these laws or a law that relates thetemperature of a gas with the pressure exerted on the gas.

Can you think of other real world settings where direct or inverse variations model therelationship between two quantities?

Investigation & Exploration (Problem #7)A stem-and-leaf plot is a quick way to picture the shape of a distribution while including the

actual numerical values in the graph. The stem of the plot is a vertical number line that representsa range of data values in a specified interval. The leaves are the numbers that are attached to the

particular stem values. In the stem-and-leaf plot for #7, the length, inmonths, of the investigations ranged from 5 to 54. The mode is 8. Thereare 54 data entries, an even number. The median is the mean of the 27th and28th entry which is (12+13)÷2 = 12.5. This is a lot of information that can beinterpreted from a single representation of the data. What are somesituations when using a stem-and-leaf plot is the best representation? Canyou see how a stem-and-leaf plot is easily turned into a histogram? Whenshould a histogram or pie chart be used? What are some other ways ofrepresenting data?

Answers

8. (M, F)

9. 39 (G, P)

10. 6 (M, P)

�� �

� �

��

��

��

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©MATHCOUNTS 2001�02 24

WARM-UP 121. ________ Ernie has 450 Galaxy cards. Sara has 10 more than Ernie and twice as many as Bert.

How many Galaxy cards do they have altogether?

2. ________ The points A through G are evenly spaced onthe number line. If A = 1/3 and G = 5/6 whatis the value at point C, expressed as acommon fraction?

3. ________ The number y is 125% of another number x. What percent of 8y is 5x?

4. ________ The five-digit number 4a,ab7 is divisible by nine where a and b are single digit wholenumbers. How many possible combinations are there for a and b?

5. ________ How many integers in the range 500 to 999 have no consecutive identical digits? Forexample, 626 is an integer with no consecutive identical digits, but 722 is not.

6. ________ In a coordinate plane, point A (4, -2), is reflected over the x-axis and labeled A�.A� is reflected over the y-axis and labeled A�. What is the sum of the coordinates ofpoint A�?

7. ________ A rectangle has dimensions 20 inches by 30 inches. If one side is increased by 30%and the other side is decreased by 20%, then what is the largest possible number ofsquare inches in the area of the new rectangle?

8. ________ Three friends, Ralph, Emerson and Waldo, each select a number from the set{1,2,3,11,12,13,21,22,23} and remove it. Then they add their three numbers together.If they put their numbers back and repeat this process for all possible combinations,how many different sums can they get?

9. ________ ABCD is a rectangle, AB = 6, BC = 4, EFGH is aparallelogram, AE/BE=2/1, and BF/FC = 1/3. What is theratio of the area of parallelogram EFGH to the area ofrectangle ABCD? Express your answer as a commonfraction.

10. _______ The sequence 2,3,5,6,7,8,10,11 � consists of the positive integers which are notperfect squares. What is the 100th member of the sequence?

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©MATHCOUNTS 2001�02 25

WARM-UP 12

1. 1140 (C)

2. 1/2 (M, F)

3. 50 (F, G)

4. 11 (P, T)

5. 405 (P)

6. -2 (M)

7. 624 (F)

Solution � Problem #3Direct translation of words into math symbols is a good approach to this problem. With this

kind of problem, we can remember that the word �is� translates into �=� and the word �of�translates into multiplication.

I. The number y is 125% of another number x. II. What percent of 8y is 5x? y = 1.25(x) ? (8y) = 5x

We will be able to manipulate the second sentence by using the first sentence. From sentence I,we know that we can substitute 1.25x for y when we work with sentence II. Therefore sentence IIbecomes: ? (8(1.25x) = 5x . This can be simplified to: ? (10x) = 5x . And this will bring us to ourfinal answer... ? = (5x)/(10x) = 1/2 = 50%.

Solution � Problem #4A number is divisible by nine if the sum of its digits is divisible by nine. The sum of the digits of

4a,ab7 is 4 + a + a + b + 7 = 2a + b + 11. The expression 2a + b + 11 must be equal to 18, 27 or 36 inorder for 2a + b + 11 to be divisible by 9.

(Why can�t 2a + b + 11 be equal to 0, 9, 45, 54, 63, . . .?)

2a + b + 11 = 18 2a + b + 11 = 27 2a + b + 11 = 362a + b = 7 2a + b = 16 2a + b = 25

If a = 0, b = 7 If a = 4, b = 8 If a = 8, b = 9a = 1, b = 5 a = 5, b = 6 a = 9, b = 7a = 2, b = 3 a = 6, b = 4a = 3, b = 1 a = 7, b = 2

a = 8, b = 0

There are 11 combinations of a and b that will make 4a,ab7 divisible by 9.

Investigation & Exploration (Problem #4)If graphed on the Cartesian coordinate grid, where x = a and y = b, what will the scatter plot of

the 11 combinations look like? Investigate the connection between the graph and the solution. Canyou use what you just discovered to find all of the possible values for c and d to make 1d,3cc,dc4divisible by 9? Though the graphing method may not be the quickest way to solve this problem, itbrings up many interesting questions and connections. How is this similar to or different from theoriginal problem in the Warm-Up?

Answers

8. 29 (P, T)

9. (F)

10. 110 (P, E)

��

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©MATHCOUNTS 2001�02 26

WARM-UP 131. ________ In March 2000, demolition experts used 4461 pounds of gelatin dynamite to bring

down the Kingdome, a stadium in Seattle. The dynamite was stuffed into 5905 holesthroughout the dome. What was the mean number of ounces of dynamite per hole?Round your answer to the nearest whole number.

2. ________ How many squares are determined by the grid linesto the right if the 48 smaller quadrilaterals arecongruent squares?

3. ________ The population of the United States is 275 million people. The land area of theUnited States is 3.6 million square miles. If the land area of the United States wereequally divided among all of its population, each person would �own� a square piece ofland K feet by K feet. Calculate K to the nearest foot.

4. ________ Two years ago the ages of an oak tree and that of a younger oak tree were bothperfect squares. Two years from now both of their ages will be perfect cubes.What is the sum, in years, of the current ages of the two oak trees?

5. ________ Miami-Dade County, Florida school officials reported 84,000 students attendingclasses in portable classrooms in the fall of 2000. If the mean number of studentsper portable classroom is 30, how many portable classrooms were being used?

6. ________ The product of the digits of a four-digit number is 6!. What is the smallest possiblevalue of this number?

7. ________ Two standard 6-sided dice are tossed. What is the probability that the sum of thetwo numbers rolled is greater than nine? Express your answer as a common fraction.

8. ________ Al, Ed and Tom are different ages. Exactly one of the following three statements istrue: I.Ed is the oldest. II.Al is not the oldest. III.Tom is not the youngest.Who is the youngest boy?

9. ________ What is the sum of the coordinates of the point on the line x + y = 4 closest to theorigin?

10. _______ Square ACEG is inscribed in the regular octagon ABCDEFGH asshown. What is the ratio of the area of square ACEG to thearea of octagon ABCDEFGH? Express your answer as a fractionin simplest radical form.

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©MATHCOUNTS 2001�02 27

WARM-UP 13

5. 2800 (C)

6. 2589 (P, G, E)

7. 1/6 (P, T)

Solution � Problem #2For this kind of counting problem, it�s best to

have a plan for how you will count each square.Probably the easiest way will be to count the numberof squares of a given size. We�ll start with thesmallest, a 1x1 square:1x1 squares (shown in the first diagram) 482x2 squares (first diagram) 243x3 squares 04x4 squares (first diagram) 15x5 squares (shown in second diagram) 46x6 squares (second diagram) 97x7 squares (shown in third diagram) 48x8 squares (third diagram) 1

A grand total of 91 squares!

Solution � Problem #8Analyze each of these three possibilities (A,B and C):

A B CStatement I T F FStatement II F T FStatement III F F T

Possibility A: Since II, �Al is not the oldest,� is false, we know that Al is the oldest. But thiscontradicts the true statement, I, �Ed is the oldest.� Possibility A is impossible.

Possibility B: Since III, �Tom is not the youngest,� is false, Tom is the youngest. However,since �Ed is the oldest � is false, while �Al is not the oldest � is true, no one can be the oldest.Possibility B is impossible.

Possibility C: Since II, �Al is not the oldest,� is false, Al is the oldest. Since III, �Tom is notthe youngest,� is true, only Ed can be the youngest. Ed being the youngest is allowed since I, �Ed isthe oldest,� is false.

Connection to... Social Studies (Problem #3)The population density of countries varies dramatically and can be represented in many ways.

The image of a 609 foot by 609 foot square for each United States resident is quite differentfrom a 1894 foot by 1894 foot square for each Canadian or a 43 foot by 43 foot square for eachresident of Singapore.

At the website http://www.undp.org/popin/wdtrends/6billion/t09.htm, the 1999 populationdensities for all countries is given in the unit �people per square kilometer.� For example, the UnitedStates of America has 29 people per square kilometer. A challenging arithmetic problem is toconvert between �people per square kilometer� and �N foot by N foot square per person.� Can youdevelop a formula for this conversion?

Answers

8. Ed (T)

9. 4 (M, G)

10. (F)

1. 12 (F, C)

2. 91 (P, S)

3. 604 ( C)

4. 129 (G, T)�

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©MATHCOUNTS 2001�02 28

WARM-UP 141. ________ The point (2,3) is reflected over the x-axis to point P. Then P is reflected over the

line y=x to point Q. What is the x-coordinate of Q?

2. ________ What is the largest value of n for which 2n divides 10! ?

3. ________ It was reported that California�s non-Hispanic white population was 57 percent ofthe total California population in 1990 and 49.8 percent in 1999. There were 17million non-Hispanic whites in California in 1990 and 16.5 million in 1999. By how manypeople did California�s population increase from 1990 to 1999? Express your answerto the nearest million.

4. ________ The numbers 1,2,3,4,5 and 6 are placed in the six circles shown tothe right so that the sum along each edge of the triangle is thesame. What is the smallest possible sum along an edge?

5. ________ Given six distinct points on a line, how many distinct segments can be named usingthe six points?

6. ________ What is the area, in square units, of the largest regular hexagon which can beinscribed in an equilateral triangle with area 24 square units?

7. ________ A customer purchased a large circular pizza with a 12-inch diameter. When shebegan eating the pizza at home, she found an unusual amount of crust on the outeredges with no sauce, cheese or toppings. She measured the outer edge in a numberof places and found it to be a 2-inch border all the way around the pizza. Whatpercent of the pizza was without sauce, cheese or toppings? Round your answer tothe nearest whole number.

8. ________ What is the sum of all prime numbers between 35 and 70 which, when divided by 12,leave a remainder that is a prime number?

9. ________ A point is randomly selected from within the rectangle having vertices at (0,0), (2,0),(2,3) and (0,3). What is the probability that the x-coordinate of the point is lessthan the y-coordinate? Express your answer as a common fraction.

10. _______ A circle has a radius of 50 meters. The radius is increased by 40%. By whatpercent is the area increased?

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©MATHCOUNTS 2001�02 29

WARM-UP 14

5. 15 (F, M)

6. 16 (F, M)

7. 56 (F, M)

Solution � Problem #9Graphing the information in this problem will help us to visualize what we

are looking for. It will probably help us to avoid a common mistake as well...it�simportant to remember that we are not only considering integer values of x andy!

If we are trying to find the points where the x-coordinates are less thanthe y-coordinates, we are working with the points in region I.

The problem states we are able to choose any point from therectangle and then must see what the probability is that it will beinside region I. We can figure this out by using the area of theregions. The rectangle is 2x3 and has an area of 6 square units.The area of region II may be easier to find than region I since it is a righttriangle with legs of 2 units each. Its area is 2 square units. That meansthat the area of region I is 6-2 or 4 square units. So the probability ofbeing inside region I when already inside the rectangle is 4/6, or 2/3.

Connection to... Number games (Problem #4)How many different ways can you arrange these integers to get a sum of 9? Some triangles

have the numbers in the same relative positions, but are obtained by rotating or reflecting thetriangle. How many of these are there? Another way to get new arrangements is what we call the�dual� where every number is moved to the opposite side of the triangle, so numbers on the edgesbecome numbers on the corners, and vice versa.

There are many number games where numbers are arranged in patterns according to their sumsor products. Most students are familiar with Magic Squares. There are many publications devotedto math puzzles. These puzzles help with critical thinking, problem solving and general numbersense...they also make math practice fun! Students can develop their own number crosswords, withmath problems as clues, and swap them with a buddy.

Investigation & Exploration (Problem #1)A Reflection is a type of transformation used in math. Not only can you reflect one point

through a line of reflection, but you can reflect entire shapes as well. Graph three points andconnect them so that they form a triangle. If you reflect those three points over the x-axis, whatdo you notice about the coordinates of the new points compared to the coordinates of the originalpoints? Take this new triangle and now reflect the points across the line y=x. Again, notice how thecoordinates change. Does the size or shape of the triangle change? Where do you think a pointreflects that is located on the line of reflection? Given the pattern of the changes that occurredwith the coordinates of the three vertices of the triangle, can you determine, without graphing it,where the point (-3, 5) would end up if you were to reflect it over the x-axis and then over the liney=x? Write instructions to a friend telling her how she can determine, without graphing, where thepoint (6, -3) will end up if she reflects it over the line y=x and then over the y-axis.

Answers8. 310 (E, G, T)

9. 2/3 (F, M)

10. 96 (F, C, M)

1. -3 (M)

2. 8 (P, E, S)

3. 3,000,000 (F, C)

4. 9 (G)

I

II

y = x

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©MATHCOUNTS 2001�02 30

WARM-UP 151. ________ Two lines defined by the equations y=mx+4 and y=3x+b, where m and b are constants,

intersect at the point (6,10). What is the value of b+m?

2. ________ If the greatest common divisor of a and b is 100, what is the greatest commondivisor of 3a2 and 3b2?

3. ________ Jabbar wants to choose a height for his son�s basketball hoop that is in the sameproportion to his son�s height as the standard 10 foot hoop is to the average 6 foot6 inch professional basketball player. His son is 4 feet 4 inches tall. What is thenumber of inches in the height at which he should place the basket?

4. ________ P and Q are positive prime numbers with P > Q and P + Q = 124. What is thesmallest possible value of P � Q?

5. ________ What is the least natural number greater than 7 that has a remainder of 7 whendivided by 24 and also has a remainder of 7 when divided by 32?

6. ________ The eight corners of a cube are snipped off to form a polyhedronwith 6 octagonal and 8 triangular faces. What is the fewest numberof colors that can be used to paint the faces so that no pair ofadjacent faces are the same color?

7. ________ A store offers customers a card with 5 circles, each hiding a percent ofdiscount: 50%, 50%, 25%, 10%, 5%. The customer selects two circles to uncover andreceives a discount equal to the average of the two values. What is the probabilityof receiving a 50% discount? Express your answer as a common fraction.

8. ________ What is the ratio of the area of a squareinscribed in a semicircle with radius 10 inchesto the area of a square inscribed in a circlewith radius 10 inches? Express your answer asa common fraction.

9. ________ There are 67 people in a tennis tournament. A player is eliminated from thetournament after losing 4 matches. What is the maximum number of matches thatcan be played so that exactly 5 people are left in the tournament?

10. _______ A circle, a square and a triangle are all drawn in a plane. None of the square�s sidesare collinear with the sides of the triangle. What is the largest number of pointsthat can belong to at least two of the three figures?

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©MATHCOUNTS 2001�02 31

WARM-UP 15

5. 103 (G, E)

6. 4 (M, G)

7. 1/10 (T)

Solution � Problem #8To find the area of the square inscribed in the

semicircle, let x equal one-half the length of the side of thesquare. Then 2x equals the length of the side of the square.We can then use the Pythagorean Theorem on the righttriangle we form by inserting the dotted line shown (a radius ofthe circle). By the Pythagorean Theorem, we have .

Solving for x, we get . The area of this square is square inches.

To find the area of the square inscribed in the circle, we can use the formula for finding thearea of a rhombus which uses its diagonal measures, or square inches.

The problem asks for the ratio of the area of the square inscribed in the semicircle to the areaof the square inscribed in the circle, so we have (80)/(200) or 2/5.

Connection to... Sports (Problem #3)Many times accommodations are made in sports for younger athletes. Steps are taken to ensure

that the challenge of the game is not too overwhelming because of a youngster�s smaller size. Themost common change is in the size of the playing area; smaller basketball courts, baseball fields andsoccer fields are used for younger players. Other times the size of the equipment is altered or thelength of playing time is shortened. In the game of tennis, however, the court size is not alteredand the size of tennis balls remains the same. Suppose you would like to adapt the game of tennisfor a younger player. Research the dimensions of the standard tennis court, the height of the netand the diameter of the ball. Assume these measurements are good for a 6 foot adult. What wouldthe new measurements be if you were to change them proportionally for a 4 foot 2 inch boy?Compare the ratio of change for the height of the net, the surface area of the court and thevolume of the ball. Do you see a connection?

Investigation & Exploration (Problem #6)One of the most famous mathematical theorems proven in the 20th century was the �Four Color

Theorem�. This theorem establishes that any map in the plane can be colored with four colors, sothat no adjacent regions have the same color. Of course, some maps can be colored with only three,or even two colors. How many colors are needed to color a cube so that adjacent faces do not sharethe same color? A tetrahedron? Try drawing a map in the plane which cannot be colored with fewerthan four colors. The history of this theorem is very interesting, with many failed attempts atproving it. Not until the 1970�s, with the aid of a computer program taking 1200 hours to run, didmany people believe an actual proof had been found!

Answers8. 2/5 (F, C)

9. 263 (T, P)

10. 20 (M)

1. -7 (M, C, F)

2. 30,000 (P, S)

3. 80 (C, F)

4. 18 (E, G)

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©MATHCOUNTS 2001�02 32

WARM-UP 161. ________ Palavi�s score on a quiz was 78, which was 130% of Samidh�s score. What was

Samidh�s score?

2. ________ When purchasing 3 hamburgers at $2.40 each at the local restaurant, the customerreceives an additional hamburger free of charge. Four people evenly share the costof four hamburgers. What is the number of dollars, to the nearest hundredth, ineach person�s share?

3. ________ The puzzle pictured consists of three identical cubes which may betwisted until each vertical face is a single color. What color is theface marked �?� ?

4. ________ How many different whole numbers between 0 and 1000 have digitswhose sum is 9?

5. ________ Let P(n) and S(n) denote the product and the sum of the digits of the integer n,respectively. For example, P(23)=6 and S(23)=5. How many two-digit numbers Nsatisfy N=P(N)+S(N).

6. ________ Given the line y = 2x � 10, what is the length, in units, of the longest segment of thisline that lies in Quadrant IV? Express your answer in simplest radical form.

7. ________ There are eight unit squares in the plane that have two or more vertices in this2 by 3 array of lattice points. How many unit squares have at least two vertices inan m by n array of lattice points?

8. ________ Two points are randomly selected from the set of ordered pairs {(0,0), (1,0), (2,0),(0,1), (1,1), (2,1)}. What is the probability that they are one unit apart? Expressyour answer as a common fraction.

9. ________ The sum of the squares of two whole numbers is 160. The product of the numbersis 48. What is the absolute value of the difference between the two whole numbers?

10. _______ Find an integer n satisfying .Q

Q Q+

+=

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1 61 6

!

!

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!

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©MATHCOUNTS 2001�02 33

WARM-UP 16

5. 9 (P, E, G)

6. (M)

7. mn+m+n-3 (P)

Solution � Problem #10Solution 1:Expanding and canceling yields the equation .

Now break 60,480 into its prime factorization:

By observation we see that we need to use the prime factors and their powers to form fourintegers that multiply to 60,480 and are almost consecutive, except that the largest will be twobigger than the next largest. Going through possible choices of combinations will not take long.Trying to use the 5 and 7, we see that the product of 5,6,7 and 9 is not large enough. If we multiplythe 7 by a 2, then we must also make the 5 larger to keep them within a five integer range of eachother. But if we multiply the 5 by a 2, this leaves 10, _ , _ , and 14 and there isn�t any way to makean 11 out of the prime factorization integers left. If we try to multiply the 5 by a 3 instead of the2, we end up with 15 and 14 and a good set of factors left to make 16 and 18...so we have 14,15,16and 18, all of which do multiply to 60,480. So we now know that n + 1 = 14 and n = 13.

Solution 2:Again, expand and cancel, which gets us .

Therefore, 60,480 is approximately (n + 3)4 . Thus, . Trying n = 13 yieldsa solution.

Connection to... Games (Problem #3)The puzzle Instant Insanity is played with four different cubes, each with faces that are red,

white, green or blue. The puzzler is asked to stack the four cubes so that each side of the new(vertical) box shows each of the four colors. This may seem like a simple task at first, but thereare over 40,000 combinations to try if you just go at it randomly. Check out the many websitesdedicated to this age-old game to see how the game works and to find good ways to approach findingthe solution. There are also several books about discrete mathematics that address and solve thisproblem.

Investigation & Exploration (Problem #5)Are there any three-digit numbers that satisfy this property? If so, what are they? If not,

can you write a convincing argument that no such numbers exist? What would be some good numbersto investigate?

Answers8. 7/15 (E, G)

9. 8 (E, G)

10. 13 (C)

1. 60 (C)

2. 1.80 (C)

3. Red (P, E)

4. 55 (P, E, G)

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QQ Q

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+= + + + + =

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©MATHCOUNTS 2001�02 34

WARM-UP 171. ________ According to a published report during the summer of 2000, the cost for long-term

parking per day at 40 U.S. airports ranged from $3.00 in Little Rock, AR to $18.00in Seattle, WA. For the year 2000, how many more total dollars in revenue areproduced by a long-term parking space at the Seattle airport than a long-termparking space at the Little Rock airport, if each parking space is used every day?

2. ________ A song is written in time, meaning that there are 3 beats to the measure and eachquarter note gets one beat. A half note would get 2 beats and an eighth note gets1/2 beat. A practice exercise consists of 26 eighth notes,26 quarter notes and 21 half notes played consecutively.What is the number of measures in the exercise?

3. ________ What is the y-intercept of the line that is the perpendicular bisector of thesegment joining the points (4,7) and (-4,12)? Express your answer as a commonfraction.

4. ________ Let S = # of square units in the surface area of a cube. Let V = # of cubic units inthe volume of the cube. When the side length of the cube is doubled, by whatfactor is the ratio of S to V multiplied? Express your answer as a common fraction.

5. ________ Each term in a sequence is the sum of the previous two terms. If the sequencecontains the terms a, b, c, 12, 19 and 31, in that order, what is the value of a?

6. ________ The Riverside Bed and Breakfast Inn serves three married couples breakfast attheir round table which seats exactly six people. The hostess wants to seat guestsso that no husband and wife sit next to each other and the guests alternate male andfemale. How many different arrangements are possible? A rotation of anarrangement is not considered a different arrangement.

7. ________ The diameters of two circles are 8 inches and 12 inches. The area of the smallercircle is what percent of the area of the larger circle? Express your answer to thenearest whole number.

8. ________ What is the least whole number n such that 84 divides n! ?

9. ________ Three faces of a cube are randomly selected. What is the probability that they havea common vertex? Express your answer as a common fraction.

10. _______ Triangles ABC, CDE and EFG are equilateral with AB = 1 cm andCD = 3 cm. Points B, D and F are collinear. How manycentimeters are in the length of segment EF?

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©MATHCOUNTS 2001�02 35

WARM-UP 17

5. 2 (C)

6. 2 (P, M)

7. 44 (F)

Solution � Problem #10

Extend line BF to intersect line AG at point P. Nowbecause of the angle at P and the fact that all of the

angles of the triangles are 60°, we can use the AngleAngle Postulate to see that

are all similar. Now we can set up the followingextended proportion with the length of segment

PA = x :

or

Solving the first equality, we get 3x =x +1 or x =1/2. Plugging this into the second equality andcross-multiplying, we get n =9 centimeters.

Connection to... Biology (Problem #4)Some biologists study the ratio of the surface area to the volume of living organisms. As you

saw in this problem, increasing the size of an object without changing its shape causes the surfacearea to volume ratio to decrease. For living creatures, the volume determines the amount of bodyheat produced, while the surface area determines the amount of body heat which is radiated, orlost, to the surroundings. For this reason, smaller creatures, which have greater area relative totheir volume, tend to have higher metabolisms and body temperatures. For example, mice have amuch higher metabolism than elephants.

Additionally, smaller animals tend to have a more compact spherical shape, while larger animalshave more and longer appendages, which tends to increase their surface area to volume ratio.

Investigation & Exploration (Problem #5)This is an example of a Fibonacci - like sequence, in which each term is the sum of the two

previous terms. How far backwards can you extend the sequence? What happens if you chooseother values for the final three terms? What would happen if each term is the sum of the previousthree terms?

The actual Fibonacci sequence (0,1,1,2,3,5,8,...) is extremely popular and very useful inmathematics. Its various connections to nature are amazing. It is also related to Pascal�s triangle,another formation of numbers that involves finding the sum of two previous numbers. Can you findwhere the Fibonacci sequence can be seen in Pascal�s Triangle?

Answers

8. 7 (G, E)

9. 2/5 (T,M)

10. 9 (F)

1. 5490 (C)

2. 27 (M, C)

3. (F)

4. 1/2 (F)

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©MATHCOUNTS 2001�02 36

WARM-UP 181. ________ A news source reported that 37 percent of the Asian population in the United

States (4 million people) live in California. What is the total Asian population in theUnited States? Express your answer to the nearest million.

2. ________ The sum of an integer and its square is 6 less than the square of the next greaterinteger. What is the value of the integer?

3. ________ Tyler rolls four standard 6-sided dice and finds that the product of the numbersrolled is 450. What is the sum of the numbers that were rolled?

4. ________ What integer is closest to the number of square units in the area of a triangle whosesides are 2, 3 and 3?

5. ________ With two games remaining in the baseball season, two players have nearly identicalbatting averages. McKay has 197 hits in 580 at bats. Nickels has 196 hits in 579 atbats. In the last two games McKay has 6 hits in 10 at bats and Nickelshas 5 hits in 8 at bats. What is the positive difference between theirfinal batting averages? Express your answer as a decimal to the nearestthousandth.

6. ________ A point is randomly selected from within the triangle having vertices at (0,0), (2,0)and (0,3). What is the probability that the point is within one unit of (0,0)? Expressyour answer as a common fraction in terms of .

7. ________ A pentagon P has vertices A=(0,0), B=(7,0), C=(13,8), D=(5,14) andE=(0,14). Line L passes through the origin and divides P into twoquadrilaterals with equal perimeters. What is the sum of thecoordinates of the point F where L meets segment CD? Express youranswer as a decimal to the nearest tenth.

8. ________ What is the sum of the digits of the decimal representation of ?

9. ________ What is the remainder when the product of the first five primes is divided by 12?

10. _______ In the grid to the right, it costs exactly$1 to move from any vertex to an adjacentvertex. How many dollars does it cost togo from point A to point B along thecheapest path?

π

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©MATHCOUNTS 2001�02 37

WARM-UP 18

5. .002 (C)

6. (E, M, C)

7. 19.8 (M)

Solution � Problem #8At first, this problem may look like it�s going to take a lot of work and a calculator with a giant

screen to show all the digits in our final number! But there�s really not a lot of calculating to bedone. We just have to find a better way to write the problem. Watch how we can transform theexpression.

Now, looking at a pattern, we can see that (101 - 1)/(9) = 1, (102 - 1)/(9) = 11, (103 - 1)/(9) = 111,...Notice how the answer is always going to be an integer made up of an amount of 1�s equal to theexponent of the 10. From this we can tell that (1022 - 1)/(9) is going to be a 22-digit number with alldigits of 1. But if we then perform the last operation of adding 1 to the number, it�s still a numberwith 22 digits, but would look like this: 1111...1112. Its digits add to 23.

Another way to approach this problem is to realize that the original numerator of the fractionwill be a 23-digit number that starts with 1, has 21 zeros and ends in 8. When we divide this numberby 9, we get a 22-digit quotient with 21 ones and a final 2. The sum of these digits is 21�1+2 = 23.

Connection to... Baseball (Problem #5)In this problem we are looking at calculating batting averages for baseball. Baseball is a game

that uses many different statistics to determine how well a player is doing throughout the season.Not only is a player�s batting average used to determine his effectiveness on offense, but also hison-base percentage and his slugging percentage. What are the differences between how thesethree numbers are calculated for a player?

The position of pitcher has its own set of statistics. The primary statistic is a pitcher�s earnedrun average (ERA). Other numbers that are often considered when determining how well a pitcher isperforming are his strike-outs per inning as well as pitches per inning. The next time you watch atelevised baseball game, notice how many references are made to math-related facts!

Investigation & Exploration (Problem #9)What is the remainder when the product of the first six primes is divided by 12? Based on your

answers to this question and problem #9, develop a conjecture and prove it or find a counter-example. What if we changed the problem to ask, �What is the remainder when the product of thefirst six primes is divided by 9�? Can you find an exact answer or narrow it down to a limitednumber of possibilities?

Answers

8. 23 (P)

9. 6 (C)

10. 10 (P, M, G)

1. 11,000,000 (C)

2. 5 (C)

3. 19 (C, E, G)

4. 3 (F, M)

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©MATHCOUNTS 2001�02 1

2001-2002 MATHCOUNTS School Handbook

WORKOUTS

The Workouts consist of multi-step problems that often require students to use several pieces of theirmathematical knowledge. These problems can be used in the classroom to challenge students and toextend their thinking. The Workouts can be used to prepare students for the Target and Team Rounds ofcompetition.

Answers to the Workouts include one-letter codes, in parentheses, indicating appropriate problem-solvingstrategies. However, students should be encouraged to find alternative methods of solving the problems;their methods may be better than the one provided! The following strategies are used: C (Compute),F (Formula), M (Model/Diagram), T (Table/Chart/List), G (Guess & Check), S (Simpler Case),E (Eliminate) and P (Patterns).

MATHCOUNTS Symbols and Notation

Standard abbreviations have been used for units of measure. Complete words or symbols are alsoacceptable. Square units or cube units may be expressed as units2 or units3.

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©MATHCOUNTS 2001�02 2

WORKOUT 11. ________ If x is 150% of y, what percent of 3x is 4y? Express your answer to the nearest

whole number.

2. ________ The ratio of the number of cans of cola soda to lemon-lime soda to cherry sodaconsumed at a graduation party was 12:3:10. If a total of 150 cans of these threeflavors of soda were consumed, how many cans were lemon-lime soda?

(For #3 - #5) The graph depicts the schedule of three trains from Washington, DC to Philadelphia.The y-axis indicates the distance in kilometers of each city from Washington, DC.

3. ________ Tomas left DC at 8:10 am,stopped to shop in XXX and thencaught the next train toPhiladelphia (Phil). How manytotal minutes did Tomas spendriding on the trains?

4. ________ What is the average speed, inkilometers per hour, of the trainfrom DC to Philadelphia thatleaves DC at 9:20 am? Expressyour answer as a decimal to thenearest tenth.

5. ________ What is the average speed, in kilometers per hour, of the fastest train betweenXXX and Philadelphia? Express your answer as a decimal to the nearest tenth.

6. ________ What is of 140? Express your answer as a decimal to the nearest hundredth.

7. ________ A goat is attached to an L-shaped rod with a leash thatallows the goat to move a ground distance of 8 metersfrom the rod on all sides. AB = 10 m, BC = 20 m and AB isperpendicular to BC. The attached end of the leash maymove along the entire rod and the goat may roam on allsides of the rod. What is the number of square meters inthe area of the region of grass that the goat can reach?Express your answer in terms of π .

8. ________ What is the number of square units in the area of a triangle whose sidesare 5, 6 and units. Express your answer in simplest form.

9. ________ The product of a set of distinct whole numbers is 120. What is the least possiblesum of the members of the set?

10. _______ If a = 4.9 and b = , determine the value of the reciprocal of b/a . Expressyour answer as a common fraction.

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©MATHCOUNTS 2001�02 3

WORKOUT 1

1. 89 (C)

2. 18 (F, S)

3. 210 (F, T)

4. 76.4 (F, T)

5. 79.3 (F, T)

6. .28 (C)

7. (M, C)

8. 9 (F)

9. 14 (G, T)

10. (C)

Solution � Problem #7This problem is similar to the triangular sandbox/fence

problem seen in an earlier Warm-up. We need to keep in mindthat the goat�s leash will allow him to move a ground distance ofup to 8 meters from any given point on the L-shaped rod. Thefirst picture shows all of the points that are 8 meters from thesegment portions of the rods (these are the segments runningparallel to the rods on either side of them) and all of the pointsthat are 8 meters from the tips of the rods (these are thecircles at each vertex/endpoint).

To show the outline of the actual region the goat can movein, we can look at the second figure. Some of the extrasegments within the boundaries have been taken out, while someother segments have been added. Notice the goat�s roamingspace is now divided into three rectangles (I, II and III) andthree portions of circles that we can now find the area of.

Region I: 2x8 = 16 square metersRegion II: 20x16 = 320 square metersRegion III: 10x8 = 80 square metersRegion IV: (1/4) 82 = 16 square metersBoth semicircles = One circle = 82 = 64 square metersTOTAL = 416 + 80 square meters

Connection to... Uses of Graphs (Problem #5)

This graph is similar to a simplified version of train schedules used in France in the late 1800�s.All of the daily trains between Paris and Dijon, including all stops, were represented on one graph.The trains from Paris to Dijon were represented by line segments with positive slopes while thereturn trains from Dijon to Paris were represented by line segments with negative slopes. Thesteepness of the slopes indicated the average speeds of the trains. This format made it easy fortravelers to plan their day�s itinerary. (Reference: �Elementary and Intermediate Algebra�, SecondEdition, Bittinger, Ellenbergen, and Johnson, Page 151)

Investigation & Exploration (Problem #8)Let�s call a lattice triangle any triangle whose vertices have integer coordinates. Notice that the

side lengths of such a triangle need not be integers. Can you find a lattice triangle for which (a) allthree side lengths are integers, (b) exactly two side lengths are integers, (c) exactly one side lengthis an integer and (d) none of the side lengths are integers?

Answers

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I

III

IVII

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©MATHCOUNTS 2001�02 4

WORKOUT 21. ________ The cost of a new car, including n% sales tax, was $20,276.50. The cost before

sales tax was $18,950. What was the value of n ?

2. ________ Between 10:50 AM and 1:30 PM, Bill rides 42 miles on his bicycle.What is his average speed in miles per hour? Express youranswer as a decimal to the nearest hundredth.

3. ________ What is the total number of digits used when the first 2002 positive even integersare written?

4. ________ Kelly�s average score on four Spanish tests is 85.5. The average of her threehighest scores is 87, and her two lowest scores are the same as each other. What isthe average of her two highest test scores?

5. ________ A solid cube measures 21 cm on an edge. Nine cubes of edge3 cm are removed from the center of each face of theoriginal cube. What is the number of square centimeters inthe surface area of the new object?

6. ________ Benson has Golden Delicious apples, each of which weighs .6 pounds, and Jonathanapples, each of which weighs .8 pounds. He wants to make applesauce such that1/3 of the weight is from Golden Delicious apples and 2/3 of the weight is fromJonathan apples. He wants to use all 12 of his Golden Delicious apples. How manyJonathan apples should he use?

7. ________ Tyrone announces, �I just found $5.00. I now have five times more money than if Ihad lost $5.00.� How many dollars did Tyrone have before finding the $5.00?Express your answer to the nearest hundredth.

8. ________ A 5 x 8 rectangle can be rolled to form two different cylinders with differentmaximum volumes. What is the ratio of the larger volume to the smaller volume?Express your answer as a common fraction.

9. ________ A �palindrome� is a positive integer that reads the same backwards and forwards.For example, 727 and 888 are palindromes. What is the largest 4-digit palindromewhich is the sum of 2 different 3-digit palindromes?

10. _______ N is a natural number such that 2x > x8 for all x > N. What is the minimum value forN?

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WORKOUT 2

1. 7 (F, C)

2. 15.75 (F, C)

3. 7456 (P)

4. 90 (T, E, F)

5. 3294 (P, M)

6. 18 (F, C)

7. 7.50 (F)

Solution � Problem #9Let�s take what we know about addition problems and see if we can figure this puzzle out! We�ve

been given the following addition problem:

A B A + C U C E F F E

E is the �Carry Digit� from A + C, therefore, E = 1. Since we are only adding together two digits,their sum can�t be more than 19, even assuming a 1 was carried over from adding the tens column.Now we know that the units column, A + C, must equal 11.

Since A + C equals 11 in the units column, they must also equal 11 for the hundred�s column.Therefore, F = 1 or 2 (if a 1 has been carried over from the tens column). So the largest that EFFEcan be is 1221.

Many combinations yield N = 1221, for example: 787 + 434 = 1221. Can you find others?

Connection to... World Population (Problem #10)2x is an example of �exponential growth� and x8 is an example of �polynomial growth.� For all

sufficiently large values of x, exponential growth (bx , b>1) always exceeds polynomialgrowth (xk , k>1).

In the late 1700�s, the English political economist, the Reverend Thomas Malthus, averred thatworld population growth is exponential while the increase in food production is polynomial (linear, infact). He therefore concluded that the English economy would face increasing problems in trying toprovide relief for the poor. The Reverend Malthus� predictions have not come true because over thelast 200 years, food production has increased exponentially due to fertilizer, technology andimproved methods of farming. However, Malthus did provide one of the earliest examples of usingmathematical concepts to forecast the future and to address social issues. (References: http://www.cs.hmc.edu/~belgin/Population/malthus.html ; http://www.stolaf.edu/people/mckelvey/envision.dir/malthus.html )

Investigation & Exploration (Problem #8)Explore rectangles that have dimensions a x b. For example, for what values of a and b will the

ratio of the volumes be 2:1? What generalization can be made?

Answers8. 8/5 (F)

9. 1221 (E, P)

10. 44 (E, G)

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WORKOUT 31. ________ A pendulum oscillates with period, P, such that P = and m/s2 and

L is the number of meters in the length of the pendulum. What is the number ofmeters in the length of a pendulum with a period of one second? Express youranswer as a decimal to the nearest hundredth.

2. ________ Sixty-one percent of the world�s population live in Asia. Of the remainder, 14% livein South America and 49% of all South Americans live in Brazil. What percent ofthe world�s population lives in Brazil? Express your answer as a decimal to thenearest tenth.

3. ________ Calculate . Express your answer as a common fraction.

4. ________ A car traveled at an average rate of 66 feet per second for 160 minutes. Howmany miles did the car travel?

5. ________ Heron�s formula (sometimes called the semi-perimeter formula) says that if atriangle has side lengths a, b and c, then the area of the triangle is given by

A= where . As a decimal to the nearest tenth,

how many square inches are in the area of the triangle with sides of length 4, 5 and6 inches?

6. ________ What is the number of square centimeters in the area of a semicircular region witha perimeter of 20 cm? Express your answer as a decimal to the nearest tenth.

7. ________ The speed of light is 670,000,000 miles per hour. How many seconds does it takelight to travel 121,000,000 miles? Express your answer to the nearest second.

8. ________ Circles A and B are externally tangent and have radii 9 inches and16 inches, respectively. How many inches are in the length ofthe common tangent MN?

9. ________ Three dimensional tic-tac-toe is played on a 3 x 3 x 3 array of lattice points. Towin, you must choose three points which lie along the same line. How many differentways can such a line be formed?

10. _______ The proper divisors of 12 are 1, 2, 3, 4 and 6. A proper divisor of an integer N is apositive divisor of N that is less than N. What is the sum of the proper divisors ofthe sum of the proper divisors of 284?

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©MATHCOUNTS 2001�02 7

WORKOUT 3

1. .25 (F)

2. 2.7 (C, F)

3. (C)

4. 120 (F)

5. 9.9 (C, F)

6. 23.8 (F, M)

7. 650 (C, F)

Solution � Problem #9In order to count the lines, we can classify them by the location of their midpoint.� Suppose the midpoint is on an edge of the cube. There are 12 edges of the cube, each of

which gives one such line.� Suppose the midpoint is the center of a face of the cube. There are 6 such points, each of

which is the middle of 4 different lines, yielding 24 total lines.� Suppose the midpoint of the line is the centermost point of the grid. There are 26 other

points and each of the 13 pairs of opposite points defines a line, which passes through thecentermost point.

Thus, there are 12 + 24 + 13 = 49 ways such lines can be formed.

Connection to... Astronomy (Problem #7)The speed of light is so fast that in the 17th Century astronomers were unable to measure its

speed and some hypothesized light was infinitely fast. Based on an extensive study of astronomicaldata on sightings of Jupiter and its moon Io, the 21-year old Danish astronomer Ole Roemer hadformed the hypothesis that the speed of light was approximately 670 million miles per hour andthat, based on the location of Earth and Jupiter in their orbits around the sun, the light from Iowould have to travel 121 million miles farther than the last time the same measurements were made.Based on these hypotheses, in 1671 Ole Roemer predicted that the next sighting of Jupiter�s moon,Io, would be 10 minutes and 50 seconds (that is, 650 seconds) later than predicted by the prominentastronomer Cassini. Even though Roemer�s prediction was proven correct, eminent astronomersrefused to accept his hypotheses until another 50 years had passed. (Reference: �A Biography ofthe World�s Most Famous Equation � E = mc2 � by David Bolanis.)

Investigation & Exploration (Problem #10)

The numbers 220 and 284 are called an amicable pair of numbers. Each is the sum of the properdivisors of the other. Show that 1184 and 1210 is also an amicable pair. Next try this method forgenerating amicable pairs.

Let a = 3�2n - 1, b = 3�2n-1 - 1 and c = 9�22n-1 -1. Suppose all three of a, b and c are prime numbers.Then the pair 2n�a�b and 2n�c is an amicable pair. For n=4 we get that a=47, b=23 and c=1151, whichare all primes, so 24�23�47 and 24�1151 is an amicable pair. Check this out by finding the sum oftheir proper divisors.

Answers

8. 24 (F, M)

9. 49 (P)

10. 284 (P, E)��� ����

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©MATHCOUNTS 2001�02 8

WORKOUT 41. ________ Miss Johnson has m guests over for a cookout. She makes 24 hamburgers, and all

guests receive the same number of burgers. How many possible values are there form if Miss Johnson has more than one guest over?

2. ________ The sum of 40 consecutive integers is 100. What is the largest of these 40integers?

3. ________ What is the largest integer k such that k4 < 106 ?

4. ________ The UPC code, made up of numbers and dashes, on a video tape is 9-78094-11006-x.The digit x is in the 12th position. Let n be the value obtained by adding the digits inthe odd positions, tripling that sum, and then adding the digits in the even positions.A valid UPC code is such that n is divisible by 10. For what value of x will this UPCcode be valid?

5. ________ The product of three consecutive positive integers is 10,626. What is the sum ofthe three integers?

6. ________ City Cab Company charges $1.60 plus $0.25 per 1/8 mile. The distancefrom the airport to the Ritz hotel is 13.25 miles. Two passengerswill share the fare equally. How many dollars will each passengerowe? Express your answer to the nearest hundredth.

7. ________ Points A(12,0), B(0,16) and C(10,10) are connected to form a triangle. From the sixpoints determined by A, B, C and the midpoints of each of the sides of the triangle,what is the number of units in the shortest distance from any of these six points tothe origin?

8. ________ Dot has a bag of apples and each apple weighs exactly 0.7 pounds. When the applesare placed on the scale, the scale shows .3 pounds, because, while the tenths digitlights up correctly, the lights recording the number of pounds are malfunctioning.What is the fewest number of apples that could be in the bag?

9. ________ Two horses on a merry-go-round are placed 8 and 17 feet from the center of thecircular path they follow. The horses make one complete rotation in nine seconds.What is the positive difference, in feet per second, between the average speeds ofthe horses? Express your answer as a decimal to the nearest tenth.

10. _______ Wheels A, B and C are attached bybelts as shown, and the two partsof wheel B are connected and turntogether as one wheel. The radiiof the two larger wheels are 6inches and the radii of the twosmaller wheels are 1 inch. Howmany revolutions will wheel C makewhile wheel A makes one revolution?

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©MATHCOUNTS 2001�02 9

WORKOUT 4

5. 66 (C, E, G)

6. 14.05 (C)

7. 10 (F, M)

Solution � Problem #1The question really asks for the number of positive integer divisors of 24. We could list the

possibilities: 1, 2, 3, 4, 6, 8, 12 and 24. Remember, we know she does not have just one guest, sothere are 7 possible values for m. For numbers larger than 24, though, listing the factors could becumbersome. It would be nice to have a method for counting the divisors of an integer, withouthaving to list all of them. Each divisor is the product of some of the prime factors of 24, so beginby prime factoring 24 as 23×3. Then, each divisor is of the form 2a×3b, where a is either 0, 1, 2, or3 (4 possibilities) and b is either 0 or 1 (2 possibilities). This implies that there are 4�2=8 factorsof 24. The factors are shown in the table below:

20 21 22 23

30 1 2 4 831 3 6 12 24

Connection to... UPC codes (Problem #4)Universal Product Codes (UPCs) are constructed so that the first digit represents the type of

product, the next five digits identify the manufacturer, the five after that label the specificproduct and the final digit is a �check digit.� A computer scanner can then make the computationdescribed in this problem. If the result isn�t divisible by 10, the computer knows that it has scannedthe numbers incorrectly. Test the UPC code on some products around your home or school.

This type of check digit is also designed to detect transpositions, or the switching of twoadjacent digits. Try swapping two digits to see if the number still satisfies the checking criteria.

Investigation & Exploration (Problem #8)This problem does not focus on whole numbers of pounds. Instead it focuses on the

remainders. This type of arithmetic is called �modular� arithmetic; in this problem, we are usingmod 10. The equation we�re trying to solve, then, is 7n = 3 (mod 10). In regular algebra, we�d solvean equation like this by multiplying by (1/7). In mod 10 arithmetic, though, that would give us ananswer of 3/7, which isn�t really what we mean. Instead, notice that 3�7 is 21, which has a unitsdigit of 1, or a remainder of 1 when divided by 10. This means that 21 = 1 in mod 10. Thus,multiplying both sides of the equation by 3 yields 21n = n = 3�3 = 9, so n is 9.

Answers8. 9 (P, T)

9. 6.3 (F, M)

10. 36 (P, F, C)

1. 7 (P, T, C)

2. 22 (P, C, G)

3. 31 (E, C, G)

4. 9 (C)

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©MATHCOUNTS 2001�02 10

WORKOUT 51. ________ Four thousand young people are attending a rock concert. Five percent are wearing

exactly one earring. Fifty percent of the other ninety-five percent are wearingexactly two earrings. Ten percent of the remaining people are wearing exactly threeearrings. The rest of the people are not wearing any earrings. What is the totalnumber of earrings being worn at the concert?

2. ________ How many three-digit numbers contain the digit 3 at least once?

3. ________ The hypotenuse of a right triangle is twice the length of one leg of the triangle. Thelength of the other leg is 12 cm. How many square centimeters are in the area ofthe right triangle? Express your answer in simplest radical form.

4. ________ Using each of the digits 2 through 9, one per square, what is the maximum value ofthe following expression?

5. ________ A square and an equilateral triangle are inscribed in a circle. What is the ratio ofthe area of the triangle to the area of the square? Express your answer as afraction in simplest radical form.

6. ________ Dora�s Delicious Doughnuts made its first batch of doughnuts oneMonday morning at 8am, and has continued to make freshdoughnuts every five hours ever since. How many weeks will it bebefore Dora�s Delicious Doughnuts makes fresh doughnuts on aMonday morning at 8am again?

7. ________ What integer is closest to the value of ?

8. ________ What is x, if x12 = 2? Express your answer as a decimal to the nearest hundredth.

9. ________ A circular table is tangent to two adjacent walls of arectangular room. Point P, on the edge of the table, is12 inches from one wall and 16 inches from the other wall asshown to the right. What is the number of inches in thediameter of the table? Express your answer to the nearestwhole number.

10. _______ Four standard 6-sided dice are rolled. The product of the four numbers rolled is144. How many different sums of four such numbers are possible?

� �� �� +

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©MATHCOUNTS 2001�02 11

WORKOUT 5

5. (F, M)

6. 5 (P, F)

7. 9 (C)

Solution � Problem #9In the diagram we can see where P is 12� and 16� from the walls,

and three radii have been drawn in, one to P and one to each of thewalls perpendicularly. There are also two secants drawn in thepicture which each run parallel to a drawn radius. Notice the righttriangle that is formed. Using the Pythagorean theorem, we seethat r2 = (r-16)2 + (r-12)2. After multiplying out each squaredbinomial, combining like terms and moving each term to the right,we have 0 = r2 - 56r + 400.

One way to solve this quadratic is with the quadratic formulawhich says:

x=

Notice that our second value of x is not possible for this drawing. Also remember that we arelooking for the diameter, so we must multiply our answer for x by 2, giving us approximately95 inches.

Solution � Problem #10The prime factorization of 144 is 24 32 = 2�2�2�2�3�3, so dice rolls of 1, 2, 3, 4 and 6 must be

considered.It is often easiest to solve counting problems by breaking them into cases:Two sixes: If two sixes are rolled, their product will be 36. So 144/36 = 4. The only ways to

get a product of 4 from two remaining dice are 2,2 or 1,4. Therefore, two sums come from 6,6,2,2and 6,6,1,4.

One six: If one six is rolled, we know 144/6 = 24. The only way to get a product of 24 fromthree dice (without using any more 6�s) is 3,4,2. The third sum is from 6,3,4,2.

No sixes: The only way to get a product of 144 without rolling any sixes is 3,3,4,4, giving us thefourth and last possible sum.

Connection to... Music (Problem #8)Each note of the musical scale is characterized by a particular frequency. The A above middle C

on a piano, for example, is 440 Hz, or cps (cycles per second). The piano keyboard has a repeatingpattern of 12 keys - 7 white and 5 black - and the ratio between every two consecutive keys/noteswas chosen so that the ratio between octaves would be exactly two. For example, the next higher Ahas frequency 880 Hz. The value of x in this problem is the ratio between two consecutive notes.Using your value of x, find the integer value of m so that xm is closest to 4/3. If you play two notesthat are m apart, their tones sound good together. Find the integer value of n so that xn is closestto 3/2. Notes that are n apart are also pleasing to the ear.

There are many ways that math ties into music. Even Pythagoras, whom we generally associatewith right triangles, did a lot of work examining the relationships between notes. Check out the�Online Math Applications� at http://tqjunior.thinkquest.org/4116/Music/music.htm for moreinformation on the many ties between math and music!

Answers

8. 1.06 (C, E)

9. 95 (M, F)

10. 4 (T)

1. 4570 (C, F)

2. 252 (T, P, E)

3. (F)

4. 15,932 (P, E, G)

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1 6 1 61 61 6 �

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©MATHCOUNTS 2001�02 12

WORKOUT 61. ________ The number is between two consecutive odd integers. What is the product of

these two integers?

2. ________ For what value of n, where n is the units digit, is 234,56n divisible by 7?

3. ________ Given: . What is the value of p ?

4. ________ A cylindrical quarter has a inch diameter and a inch height. What would bethe number of inches in the height of a coin whose volume is exactly four times thatof the given quarter and whose diameter equals inches? Express your answer as acommon fraction.

5. ________ What is the largest four-digit number, the product of whose digits is 6! ?

6. ________ What is the x-intercept of the line perpendicular to the line defined by 3x-2y=6 andwhose y-intercept is 2?

7. ________ What is the product of all integer perfect squares less than 50?

8. ________ How many four-digit numbers have the property that each of the three two-digitnumbers formed by consecutive digits is divisible either by 19 or 31?

9. ________ A circle with diameter 2 cm is centered at a vertex D of thesquare and intersects square ABCD and equilateral triangleDCE at midpoints F and G, respectively. What is the numberof square centimeters in the area of the region obtained bytaking the union of the interiors of the three figures?Express your answer as a decimal rounded to the nearesthundredth.

10. _______ The different toppings available at Conway�s Ice Cream Parlor are given below. Acustomer walks up and says, �I�d like a scoop of chocolate ice cream with any2 different wet toppings and any 3 different dry toppings. Surprise me!� How manydifferent combinations of toppings are possible for the customer�s order?

Wet Toppings Dry ToppingsCaramel M&M�sFudge Heath BarChocolate Syrup ButterfingerButterscotch Peanuts

SprinklesGummi BearsOreosNestle CrunchPecans

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©MATHCOUNTS 2001�02 13

WORKOUT 6

5. 9852 (P)

6. 3 (F)

7. 25,401,600 (C)

Solution � Problem #8First we need to know which 2-digit numbers are divisible by either 19 or 31. For 19 we have the

following options: 19, 38, 57, 76 and 95. The multiples of 31 are 31, 62 and 93. So we are lookingfor as many 4-digit numbers as we can find whose three 2-digit numbers formed by consecutivedigits are from the list 19, 31, 38, 57, 62, 76, 93 and 95.

Let�s see if we can find a number that starts with the 19. It will be in the form 19_ _. Thereare two options for the third digit since 93 and 95 are in our list, so we have 193_ and 195_. Thefirst number can be finished with either a 1 or an 8 since 31 and 38 are in our list and the secondnumber can be finished with a 7 since 57 is in the list. So we have three numbers that startwith 19: 1931, 1938 and 1957.

Start with 31_ _ --> 319_ --> 3193 or 3195.Now start with 38_ _. There are no numbers in our list that start with an 8, so we�re finished.Let�s continue the pattern. 57_ _ --> 576_ --> 576262_ _ --> Finished...no numbers starting with a 2.76_ _ --> 762_ --> Finished...no numbers starting with a 2.93_ _ --> 931_ and 938_ --> 9319 and the second one is finished...no numbers starting with an 8.95_ _ --> 957_ --> 9576

We have found a total of 8 numbers that fit the requirements.

Investigation & Exploration (Problem #7)For Problem #7, we can see that the set of all the natural number perfect squares less than 50

is {1, 4, 9, 16, 25, 36, 49} = {1�1, 2�2, 3�3, 4�4, 5�5, 6�6, 7�7}. The product of these numbers can beshown in the following form: 1�1�2�2�3�3�4�4�5�5�6�6�7�7 which is (7!)2 or (5040)2 = 25,401,600.

From this problem, can you figure out a quick way to show the product of all the natural numberperfect cubes less than 217? What would be the product of all the natural number perfect cubesless than n3 + 1?

Answers

8. 8 (P, E)

9. 7.56 (F)

10. 504 (P, S)

1. 143 (C, E, M)

2. 3 (E, P)

3. 85 (C)

4. (F)�����

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©MATHCOUNTS 2001�02 14

WORKOUT 71. ________ One-fourth of Holttown High School�s students are seniors, one-third are juniors,

and the other 300 students are sophomores. Of the seniors, two-fifths are boys.How many senior girls are students at Holttown High School?

Hole Par TigerNumber Score Woods

10 5 411 3 312 4 313 4 414 3 215 4 416 4 417 4 3

18 5 4

2. ________ In the game of golf, par is the term used to describe thenumber of shots it should take for a professional golfer toget the ball in the hole. The par scores and Tiger Woods�scores for the last 9 holes of the 2000 ProfessionalGolfers Association Championship are shown. How manyshots below par was Tiger Woods for these 9 holes?

3. ________ What is the maximum integer value of n such that 2n is a factor of 120! ?

4. ________ The five interior angle measures of a pentagon are 2x, 3x, 4x, 5x and 6x degrees.The measures of their corresponding exterior angles are a, b, c, d and e degrees,respectively. What is the value of the largest ratio: a/b, b/c, c/d, d/e or e/a ?Express your answer as a common fraction.

5. ________ Justin is reducing the number of cans of soda he consumes each day. Aftertoday, he will wait a full day before having another. Then he�ll wait two moredays, then three, and so on, extending his waiting period by one day eachtime. In how many years (to the nearest year) will he be drinking a can ofsoda only once every 60 days?

6. ________ Call a set of positive integers a �phancy set� if the product of any two integers inthe set is 1 less than a perfect square. What is the least possible value for n suchthat {4,6,n} is a phancy set?

7. ________ Three 3-digit numbers are formed using the digits 1 through 9 exactly once each.The hundreds digit of the first number is 1. The tens digit of the second number is8. The units digit of the third number is 5. The ratio of the first number to thesecond number to the third number is 1:3:5, respectively. What is the sum of thethree numbers?

8. ________ What is the maximum number of 3� x 4� rectangles that will fit, without overlap,within a 20� x 20� square?

9. ________ The product of the digits of a four-digit number is 6!. How many such 4-digitnumbers are there?

10. _______ The following table indicates the fuel consumption, in gallons/hour, for a cartraveling at various speeds. At which of these speeds, in miles per hour, does thecar consume the least gallons of gasoline per mile?

speed (miles/hour) 10 20 30 40 50 60gasoline used (gallons/hour) .90 1.20 1.40 1.70 2.00 2.50

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©MATHCOUNTS 2001�02 15

WORKOUT 7

5. 5 (P)

6. 20 (E)

7. 1161 (P, E)

Solution � Problem #8One might start by recognizing that 20 is 5x4 so

5 rows of rectangles could be arranged in 6 three-inchcolumns using 30 rectangles. This would leave a2x20 inch strip left over. Recognizing that 3+3+2 = 8suggests that it would be possible to rotate the last tworows of rectangles and fit in more. This arrangementuses 32 rectangles with a 2x8 rectangular strip leftover. This strip has an area of 16 squares which is 4more than the area of the 3x4 rectangle. Is it possibleto rearrange the rectangles to use the remaining area?Consider a 12x8 rectangle made from eight 3x4rectangles and arrange them as we did in the figure tothe right.

This leaves a 4x4 square in the middle where onemore 3x4 rectangle may be placed, using a total of 33rectangles!

Solution and Investigation & Exploration (Problem #3)The number 120! can be thought of as being the string of factors 1�2�3�4�...�119�120.Sixty of these factors are divisible by 2. (Giving us 60 factors of 2.)Thirty of these factors are divisible by 22. (Giving us another 30 factors of 2.)Fifteen of these factors are divisible by 23. (Giving us another 15 factors of 2.)Seven of these factors are divisible by 24. (Giving us another 7 factors of 2.)Three of these factors are divisible by 25. (Giving us another 3 factors of 2.)One of these factors is divisible by 26. (Giving us another factor of 2.)None of these factors are divisible by 27.Thus 60 + 30 + 15 + 7 + 3 + 1 = 116 is the total number of two factors of 120!

What if the question was changed to: What is the maximum integer value of n such that 5n is afactor of 120! ? Or 10n? How many trailing zeros are there if 120! is multiplied out?

Connection to... Golf (Problem #2)Golf is a sport in which positive and negative numbers play a role. The term, par, is the number

of strokes a skillful player is expected to take to get the ball into the hole. If the par for aparticular hole is 5, and a player gets the ball in the hole in just 3 shots, the score for the holewould be �2 because it was made in two fewer shots than expected. If the player makes it in6 shots, the score for the hole would be +1 because it was made in one more shot than expected.There are special names for some scores such as bogey, birdie and eagle. You might like to look uptheir meanings.

Answers8. 33 (P, M)

9. 72 (P, E)

10. 50 (F)

1. 108 (C)

2. 5 (C)

3. 116 (P)

4. 5/2 (F)

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©MATHCOUNTS 2001�02 16

WORKOUT 81. ________ The product ab = 1200 and b is an odd number. What is the largest possible value

of b?

2. ________ A flat steel bridge is built from two rigid 250 foot long beams joined at the middle.On a hot day, the beams expand equallycausing the joint to rise 5 feet. Byhow many inches did one of thebeams expand? Express your answeras a decimal to the nearest tenth.

3. ________ An unlimited number of darts are to be thrown at a dartboard with possible scores as shown to the left. What is thegreatest whole number score that is not possible to achieve?

(For #4 - #6) The graph represents the cost C, in dollars, of a taxi ride of distance x, in miles.

4. ________ A sign advertises the cost of a ride as having an initial fee of $a, plus $b per mile.Calculate b. Express your answer as a decimal to the nearest hundredth.

5. ________ If the graph continues as a straight line, what is the number of dollars in the cost ofa 35-mile ride?

6. ________ If the graph continues as a straight line, what is the number of miles in the length ofa ride that costs $53.60?

7. ________ Define the function a @ b = a(b) + b. What is the value of 1@(1@(1@(1@(1@1))))?

8. ________ The points A, B and C lie in a plane and have coordinates (6,5), (2,1) and (0,k),respectively. What value of k makes the sum of the lengths of segments AC and BCthe least possible value?

9. ________ What is the least whole number with exactly eleven factors?

10. _______ Find all 6-digit multiples of 22 of the form 5d5,22e where d and e are digits. Whatis the maximum value of d?

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WORKOUT 8

5. 44 (C)

6. 43 (C)

7. 32 (C)

Solution � Problem #8Because the ordered pair for point C is in the form (0,k), we know

that the point must be somewhere along the y-axis. The firstdiagram shows the segments for five different placements of C. Weneed to find the pair of segments with the shortest combineddistance.

The problem is equivalent to one obtained by reflecting (2,1) inthe y-axis (as shown in the second diagram). We can see that theshortest path from (-2,1) to (6,5) is a straight line.

We can find the slope of this line since we know two points on theline determine the slope, which is given by the following formula:

Using the slope intercept form of a linear equation, we have

y=(1/2) x + k. Substituting (-2,1) for x and y, we get 1= (1/2)(-2) + k,

and k = 2. So point C, found in the box in the second diagram, is (0,2).

Connection to... Engineering (Problem #2)Engineers need to know how different materials will expand and contract as a function of

temperature when they design buildings and highways. Investigate how different materials changeas a function of temperature. Investigate how long bridge spans can be without creating gaps thatare too large for a vehicle to pass over.

Investigation & Exploration (Problem #7)In general, what does 1@n do to n? What�s another way to write 1@1@�@1 n times? What

about 2@n? Does the position of the parentheses affect the outcome? For instance, would we getthe same answer for (((((1@1)@1)@1)@1)@1)? If the parentheses don�t affect the answer, then wesay that the operation @ satisfies the Associative Property. Does @ satisfy the AssociativeProperty?

Answers8. 2 (E, F, M)

9. 1024 (P)

10. 8 (E, C, P)

1. 75 (E, P)

2. 0.6 (F)

3. 39 (P)

4. 1.20 (F)

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WORKOUT 91. ________ By how many degrees does the measure of an interior angle of a regular decagon

exceed the measure of an interior angle of a regular pentagon?

2. ________ A small hose fills a swimming pool in 16 hours. A large hose connected to a differentwater supply fills the same pool in 12 hours. With the pool empty, the owner turns onthe smaller hose at 8:00am. He turns on the larger hose at 10:00am. Both hoses areused from 10:00am to 3:00pm. What percent of the pool is full at 3:00pm?Express your answer to the nearest tenth.

3. ________ To test whether an integer, n, is prime, it is enough to be sure that none of theprimes less than the square root of n divide n. If you want to check that a numberbetween 900 and 950 is prime with this rule, what is the largest prime divisor youneed to test?

4. ________ A circle with diameter 2 cm is centered at a vertex D of the square andintersects square ABCD and equilateral triangle DCE at midpoints Fand G, respectively. What is the number of centimeters in theperimeter of the region obtained by taking the union of theinteriors of the three figures? Express your answer as a decimalto the nearest hundredth.

5. ________ What is the sum of all of the multiples of 3 between 100 and 200?

6. ________ For how many positive integers p does there exist a triangle with sides of length3p - 1, 3p, and p2 + 1?

7. ________ A quarter, 2.5 centimeters in diameter, isdropped randomly on the tabletop shown so thatat least half of the coin lies on the tabletop.What is the probability that the quarter lies onone of the segments: or ? Expressyour answer as a common fraction.

8. ________ What is the sum of all the elements of the two-element subsets of {1, 2, 3, 4, 5, 6} ?

9. ________ Allison has sneezed exactly one million times in her life. Because of her age therehas to have been at least one day when she sneezed at least 101 times. What is theoldest, in days, Allison could be?

10. _______ If n is an integer and 20 < 2n < 200, what is the sum of all of the possible values of n?

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WORKOUT 9

5. 4950 (P, C, F)

6. 5 (G, P)

7. (M, S)

Solution � Problem #9The solution to this problem involves a useful mathematical principle called the �Pigeon Hole

Principle� (PHP). In its simplest form, the PHP says that if you havemore pigeons than pigeon holes, then there must be some pigeon holewhich has more than one pigeon! In its more general formulation, thePHP says that if you have nm + 1 pigeons in n pigeonholes, then there issome pigeon hole with more than m pigeons. To apply the pigeon holeprinciple to this problem, suppose that Allison has never sneezed asmuch as 101 times in a day. Then she must have lived at least1,000,000 (sneezes)/100 (sneezes per day) = 10,000 days. If shesneezed fewer than 100 times in a given day, then she would have had tolive even longer! But, if she�s lived fewer than 10,000 days, then theremust have been a day when she sneezed at least 101 times. To make this conclusion, therefore, shemust have lived no more than 9,999 days.

Connection to... Computer Security (Problem #3)

Eratosthenes was a Greek mathematician (ca 284-192 B.C) well known for his �Sieve.� A sieve isthe kitchen item you use to sift flour. Eratosthenes� Sieve was a filter for prime numbers. Hebegan with all the positive integers, first eliminating all the multiples of 2, then the multiples of 3and so on. For more information about Eratosthenes� Sieve, check out the web athttp://www.math.utah.edu/~alfeld/Eratosthenes.html. One interesting application of large primenumbers is their use in many types of computer security. Look into how prime numbers are used forthis purpose.

Investigation & Exploration (Problem #10)To solve an equation, we usually want to �undo� the operations on both sides of the equation.

What operation �undoes� exponentiation? You may not be familiar with the logarithm function, butit is the function that can �undo� a base from its exponent in a problem like 10x = 100, where thebase is 10. We can rewrite this as log 100 = x. Using a calculator, note that log 100 = 2. That isbecause 102 = 100.

A property of logarithms is that the log (ab) = b(log a). If we solve our Warm-Up problem usinglogs, we have log 20 < log 2n < log 200 ... notice that we are �taking the log of� each of the threeexpressions in the inequality. Using our property of logarithms, our inequality becomeslog 20 < n(log 2) < log 200. So dividing all three expressions by log 2, we arrive at(log 20)/(log 2) < n < (log 200)/(log 2). Do these calculations on your calculator to see that thisinterval includes the integers 5, 6 and 7.

Take the equation 23 = 8. We know this to be true. So if we had 2x = 8, we would expect x toequal 3. Try solving this equation using logarithms. Start by taking the log of both sides of theequation.

Answers

8. 105 (P, C, T)

9. 9999 (S, F)

10. 18 (E)

1. 36 (F)

2. 85.4 (F, M)

3. 29 (E)

4. 11.67 (F)

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