MATHCOUNTS

1999-2000 Chapter Competition

Countdown Round

On a recent trip, Jessica drove 35 mph to a store and 40 mph on

the return. If the combined driving time was 1.5 hours, how

many miles did she drive one way?

Answer: 28 (miles)

Jana gave 4 cookies to each of her classmates. She would have needed 36 more cookies to give them 7 cookies each. How many

classmates were there?

Answer: 12 (classmates)

The letter A is worth 1 point, B is worth 2 points, C is worth 3 points, ..., Z is worth 26 points, and the values of letters are added to determine the point value of a word. What is the point value of the word SEVEN?

Answer: 65 (points)

Compute: 99.3

Answer: 729

A hollow cube has a volume of 0.008 cm3. What is the number of meters in the perimeter of the figure formed by opening and flattening the cube as shown? Express your answer as a decimal to the nearest tenth.

Answer: 2.8 (meters)

Teresa has lived one-fourth of her life in England, one-fifth in Ireland, one-third in France and 13 years in Switzerland. Assuming she lived in no other countries, how many years old is she?

Answer: 60 (years)

Beginning with January 1 in a leap

year, what is the calendar date of

the last day of the first of the

year?

13

Answer: May 1

The side length of a regular hexagon is 9 inches. What is the number of inches in the difference between the greatest and the least distance between two vertices of the hexagon?

Answer: 9 (inches)

A cashier changed a $10 bill into dimes, nickels and quarters. If each coin was used the same number of times, how many total coins were used?

Answer: 75 (coins)

If the present time is 9:00 on a 12-hour clock, what hour will it

be 1447 hours later?

Answer: 4:00 (or 4 o'clock)

A circle is inscribed in a semicircle as shown. The diameter of the circle and the radius of the semicircle are both 12 inches. What is the number of square inches in the area of the shaded region? Express your answer in terms of .

Answer: 36 (square inches)

12.5% of 420 is what percent of 210?

Answer: 25 (percent)

The average of 10 numbers is 28, and the average of 8 numbers is

100. What is the average of all 18 numbers?

Answer: 60

If x, y and z are positive integers and 2x • 3y • 5z = 27,000, what is the value of x + y + z?

Answer: 9

How many of the first 10 positive integers have reciprocals that are

repeating decimals?

Answer: 4 (integers)

Alex counted to 400 by 6's beginning with 6, and Matthew counted to 400 by 4's starting with 4. How many of the numbers counted by Alex were also counted by Matthew?

Answer: 33 (numbers)

What is the sum of the two prime numbers between 110 and 130?

Answer: 240

In an isosceles right triangle whose area is 15 square units, each leg is as long as the side of a certain square. What is the number of square units in the area of the square?

Answer: 30 (square units)

A line has an x-intercept of 4 and a y-intercept of -3. What is the slope of the line? Express your answer as a common fraction.

Answer: 34

What is the value of f (4) if f (x) = x3 + 1?

Answer: 65

The Little Pieces Tuna Company is reducing the size of the radius of its best selling can of tuna by 10%. By what percent is the volume of the can reduced?

Answer: 19 (percent)

Express

as a common fraction.

78

34

45

23

Answer: 1516

A 12-foot piece of string is cut in half, and each of the two halves is used to form an equilateral triangle. What is the sum of the number of square feet in the areas of the two triangles? Express your answer in simplest radical form.

Answer: 2 3 (square feet)

Evaluate: [4 - 3(6 - 8)-1]-1. Express your answer as a common fraction.

Answer: 211

For what value of n does

218 522 = 6.25 10n?

Answer: 20

A number is chosen from the set {1,3,5}, a second number is chosen from the set {6,8,10}, and a third number is chosen from the set {7,9,11}. How many possible sums of the three numbers chosen will be odd?

Answer: 0 (sums)

How many degrees are in the measure of an interior angle of a regular pentagon?

Answer: 108 (degrees)

Given that a * b = (ab + ba) and a b = b a, what is the value of 4 (2 * 4)?

Answer: 8

Two numbers are chosen from a set of five prime numbers. One number is used as the numerator of a fraction, and the other is used as the denominator. How many unique common fractions can be formed?

Answer: 20 (fractions)

AB with endpoints (2,3) and

(6,7) is graphed on a rectangular

coordinate plane. AB is then

reflected about the y-axis. What

is the sum of the values of the

coordinates of the midpoint of

the reflected segment?

Answer: 1

For what value of x does 318 = xx?

Answer: 9

At a party, 15 handshakes were exchanged. If each person at the party shook hands exactly once with every other person at the party, what is the number of

people who attended the party?

Answer: 6 (people)

What is the units digit of the product 383 738?

Answer: 3

Adam, Ben, Chase, David and Ed were waiting in line. Adam is between Ben and Chase. Ben is between David and Adam. Ed is also between David and Adam. Ben is between David and Ed. Who is in the middle of the line?

Answer: Ed

What is the value of 25% of 14 plus 14% of 25? Express your answer as a common fraction.

Answer: 7

What is the number of degrees in the measure of the acute angleformed between the hour hand and the minute hand at 4:30 p.m.?

Answer: 45 (degrees)

Kim travels east or south on streets from his home to school. What is the number of different paths he can travel from home to school?

Home

School

N

Answer: 6 (paths)

The vertices of quadrilateral ABCD have coordinates (-2,4), (5,4), (-2,1) and (5,1). What is the number of square units in the area of quadrilateral ABCD?

Answer: 21 (square units)

There are 80 students taking a geology class, and one-fourth of the students are boys. If 20% of the boys and 10% of the girls are taking a field trip, what percent of the class is taking the trip? Express your answer as a decimal to the nearest tenth.

Answer: 12.5 (percent)

What is the least possible whole number that can be multiplied by 200 such that the product is a perfect cube?

Answer: 5

The supplement of an angle is 25 degrees more than twice the complement of the angle. What is the number of degrees in the measure of the angle?

Answer: 25 (degrees)

Ollie Origin resides at (0,0) on a coordinate plane. If Lily’s house lies at the point on the line y = x + 2 closest to Ollie, what is the number of units from Lily’s house to Ollie? Express the answer in simplest radical form.

Answer: 2 (units)

What is f ( f ( f (3))) ?

f (n) = n2, if n is evenn + 1, if n is odd{

Answer: 256

The floor of a rectangular room measures 6 feet by 10 feet. The room is 8 feet high. What is the number of square feet in the total surface area of the walls?

Answer: 256 (square feet)

Two letters are chosen at random without replacement from the word MATHEMATICS. What is the probability that both will be vowels? Express your answer as a common fraction.

Answer: 655

The equation 12 = 3 4 is formed by four consecutive digits. Given that a, b, c and d are consecutive positive integers such that 10a + b = c d, what is the sum of the next set of integers that will create a similar equation?

Answer: 26

A car is driven 40,000 miles using four tires and a spare tire. The tires are rotated so that each tire travels the same number of miles. What is the number of miles traveled by each tire?

Answer: 32,000 (miles)

The positive difference between two positive integers is 44. The product of the same two integers is 1280. What is the sum of the integers?

Answer: 84

How many circular cookies measuring 2 inches in radius can be cut from a circle of dough measuring 6 inches in radius, assuming that dough between the cookies is not reused?

Answer: 7 (cookies)

Jeremy has 11 coins that total more than $1, but no combination of the coins equals $1. What is the least number of cents that Jeremy could have?

Answer: 107 (cents)

When of a positive fraction is

doubled and the result is

multiplied by the original

fraction, the product is .

What is the original fraction?

Express your answer as a

common fraction.

110

15

Answer: 12

In the multiplication shown, a six-digit number with hundreds digit 5 is multiplied by 7. The result is a seven-digit number with hundreds digit f. Each letter represents a digit. What is the value of the sum a + b + c + d + e + f?

abc,5de × 7 = 6,744, f 56

Answer: 31

If 6 pencils and 8 erasers cost $1.00, and 8 pencils and 6 erasers cost $1.10, how many cents is the cost of one pencil and one eraser?

Answer: 15 (cents)

Niki is a competitor in the Rocky Mountain Bike Tour. She rides6 miles uphill at an average speed of 3 mph and 6 miles downhill at an average speed of 12 mph. How many hours does it take her to complete the trip? Express your answer as a mixed numeral.

Answer: 2 (hours)12

What is the greatest possible number of days in one century?

Answer: 36,525 (days)

What number is one-half of one-tenth of one-fifth of one-half of one million?

Answer: 5000

What is 4% of 21 divided by 7% of 24? Express your answer as a common fraction.

Answer: 12

The digits of a two-digit number are reversed. The positive difference between the original number and the new number is 63. What is the greatest possible new number?

Answer: 92

Mr. Demarais randomly returned the test papers to the 13 students in his class. What is the probability that exactly 12 students received their own papers?

Answer: 0

A triangle has area 7 in2. Another triangle, similar to the first, is formed by drawing lines parallel to each side of the original triangle and through the opposite vertex. What is the number of square inches in the area of the triangle formed?

Answer: 28 (square inches)

What is the least 4-digit number divisible by 2, 3, 4, 5, 6, and 7?

Answer: 1260

How many of the first 100 positive integers are divisible by 7?

Answer: 14 (integers)

ABC ADB, AC = 4 cm, and AD = 9 cm. What is the number of centimeters in the length of AB?

Answer: 6 (centimeters)

What is the mean of the first 25 odd counting numbers?

Answer: 25

Mr. Namm baked 252 cookies, Mrs. Clancy baked 105 cookies, and Mr. Palavas baked 168 cookies. Each baker packaged them with the same number of cookies in each package. What is the greatest number of cookies that could be in each package?

Answer: 21 (cookies)

Black and white unit cubes are alternately placed to form a 5 5 5 cube as shown. How many black unit cubes are in the larger cube?

Answer: 63 (cubes)

The least common multiple of 12, 15, 20 and k is 420. What is the least possible value of k?

Answer: 7

What is the greatest integer x such that x3 2000?

Answer: 12

The LCM of a pair of whole numbers is 450, and the GCF of the numbers is 6. One of the numbers is 18. What is the other number?

Answer: 150

Jody travels from mile marker 7 to mile marker 47. At which mile marker will Jody have completed 35% of her trip?

Answer: (mile marker) 21

An exterior angle of a regular polygon has a measure of 45°. How many sides does the polygon have?

Answer: 8 (sides)

Express

5+7+9+11+13+15+17 15+21+27+33+39+45+51

as a common fraction.

Answer: 13

The sum of the lengths of the diagonals of a rhombus is 28 centimeters. Given that the area of the rhombus is 96 square centimeters, what is the number of centimeters in the length of the shorter diagonal?

Answer: 12 (centimeters)

Erica bought dinner for herself and 5 friends. Each meal was $7.95, including tax and tip. What was the number of dollars in the total cost of the dinner?

Answer: 47.70 (dollars)

A watch shows calendar dates 1 through 31 and then resets itself to 1. However, it needs to be manually adjusted for months with fewer than 31 days. What is the greatest number of days over which the watch will not have to be adjusted?

Answer: 92 (days)

A bowl contains red, green and blue marbles. The probability of drawing a red marble is . The probability of drawing a green marble is . What is the probability of drawing a blue marble? Express your answer as a common fraction.

13

17

Answer: 1121

The lengths of two legs of a right

triangle are 3 cm and 5 cm.

What is the number of

centimeters in the length of the

hypotenuse? Express your

answer in simplest radical form.

Answer: 2 2 (centimeters)

The areas of the three distinct faces of a rectangular prism are 35, 15 and 21 square centimeters. What is the number of cubic centimeters in the volume of the prism?

Answer: 105 (cubic centimeters)

A sheet of paper is folded in half, then folded in half again, and this pattern of folding in half continues. What is 25% of the number of regions into which the paper is divided after the 7th fold?

Answer: 32 (regions)