MathCounts · PDF fileMathCounts Preparation How to Excel at Middle School Math Competitions...

MathCounts Preparation How to Excel at Middle School Math Competitions

By

Huasong Yin

www.jacksonareamath.com

2014, Huasong Yin

ALL RIGHTS RESERVED

This book contains material protected under International and Federal Copyright Laws and Treaties. Any unauthorized reprint or use of this material is prohibited. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without express written permission from the author.

ISBN-13: 978-1-62890-725-4

ISBN-10: 1-62890-725-8

Printed in the United States

Print Date: 12/26/2013

Email [email protected] for any errors or mistakes

Table of Contents Chapter 1 Number Sense and Speed Calculation .................................................................................. 2

Addition ........................................................................................................................................... 2

Subtraction ...................................................................................................................................... 3

Multiplication ................................................................................................................................... 4

Division ............................................................................................................................................ 7

Mental Skills for Problem Solving ..................................................................................................... 8

Translating from Verbal to Algebra ................................................................................................. 14

Chapter 2 Exponents and Fractions .................................................................................................... 17

Exponential Notation and Properties of Exponents ........................................................................ 17

Place Values and Number Systems ................................................................................................. 21

Powers of Ten and Scientific Notation ............................................................................................ 23

Divisibility, Prime Numbers and Prime Factorization ...................................................................... 28

Fractions ........................................................................................................................................ 42

Ratio, Proportion and Percent ........................................................................................................ 51

Speed - Distance Time ................................................................................................................. 60

Chapter 3 Probability, Counting and Basic Statistics ........................................................................... 64

Sets, Subsets, Set Operations and Logic .......................................................................................... 64

Counting Techniques, Permutations and Combinations ................................................................. 72

Probability ...................................................................................................................................... 91

Mean, Median, Mode, Range and Other Means ........................................................................... 101

Chapter 4 Elementary Algebra ......................................................................................................... 110

Variables and Algebraic Expressions ............................................................................................. 110

Using (a + b)2 = a2 + 2ab + b2 ......................................................................................................... 112

Difference of Two Squares: Using (a + b)(a b) = a2 b2 .............................................................. 116

Square Roots and Simplifying Square Root Expressions ................................................................ 119

Chapter 5 Geometry ........................................................................................................................ 122

Perimeter and Area ...................................................................................................................... 122

Angles, Degree Measurement and Polygons ................................................................................ 127

Triangles ...................................................................................................................................... 134

Pythagorean Theorem and Special Right Triangles ....................................................................... 139

Chapter 1: Number Sense and Speed Calcula on --- Addi on

1

Triangles and Trapezoids .............................................................................................................. 149

Circles .......................................................................................................................................... 155

Solid Objects, Polyhedrons, Volumes and Euler Theorem ............................................................. 159

Chapter 6 Sequences........................................................................................................................ 163

Sequences and Pattern Recognition ............................................................................................. 163

The Story of Little Gauss and Arithmetic Sequences ..................................................................... 164

The Smart Pizza Eater and Geometric Sequences ......................................................................... 168

Other Sequences .......................................................................................................................... 171

Chapter 7 Equations, Inequalities and Functions .............................................................................. 180

Linear Equations........................................................................................................................... 180

Quadratic Equations ..................................................................................................................... 188

Inequalities .................................................................................................................................. 195

Functions ..................................................................................................................................... 199

Chapter 8 A Little Bit of Analytic Geometry ...................................................................................... 202

Number Lines and the Coordinate System .................................................................................... 202

Lines and Circles in a Coordinate System ...................................................................................... 213

Chapter 9 A Little Bit of Number theory ........................................................................................... 226

Repeating Decimals ...................................................................................................................... 226

Modular Arithmetic ...................................................................................................................... 230

Chapter 10 Practice Tests ................................................................................................................. 234

Some Test-Taking Tips .................................................................................................................. 234

Practice Test #1 ............................................................................................................................ 235

Practice Test #2 ............................................................................................................................ 240

Practice Test #3 ............................................................................................................................ 245

Practice Test #4 ............................................................................................................................ 250

Practice Test #5 ............................................................................................................................ 255

Solutions to Practice Test 1 .......................................................................................................... 260





Chapter 1: Number Sense and Speed Calcula on --- Addi on

2

Chapter 1 Number Sense and Speed Calculation In the MathCounts Sprint round you are given 30 questions to finish in 40 minutes. You will spend one third of the time to read and understand the questions. Some of the questions are wordy. You have about one minute in average to solve and answer each question. Most of you will not be able to finish all the questions due to time constraint. Thus speed is the key to your success in the test. You do not want to waste your time on complicated calculations. You do not want to write down calculations step by step.

In this chapter, we will review the properties of numbers and the basic techniques for speed and mental calculations.

Addition Addition is commutative and associative. This means that + aa + b = b and b + c + c = a +a + b

for any numbers a , b and c . For example: 2332 + = + , 81228122 + + = + + . As a result of the commutative and associative property, we do not have to add numbers from left to right when we are adding a bunch of numbers.

Example 1: ?312927149631 =++++++

Solu on: 319146273291312927149631 +++++++=++++++ 120=6060 40+20+30+30=

Example 2: ?= 4.76+3.14+2.63+1.45+1.86+0.37+0.24

Solu on: 4.76+3.14+2.63+1.45+1.86+0.37+0.24 1.45 + 3.14)+(1.86+2.63)+(0.37+4.76)+(0.24= 14.45=1.45+13=1.45+5+3+5=

Example 3: ?53

25

34

43

52

41

31

21

Solu on: 53

52

43

41

34

31

25

21

53

25

34

43

52

41

31

21

320

35511

353

Example 4: ?= 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

Solu on: 5+6)+(4+7)+(3+8)+(2+9)+(1= 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 145 = 5 + 10 + 10 + 10 + 10=

Example 5: ?= 6997 5996 + 4995 + 3994

Solu on: 370004-6000 + 5-5000 +6-4000 = 6997 + 5996 + 4995 + 3994

21982182200034567000600050004000

Chapter 1: Number Sense and Speed Calcula on --- Subtrac on

3

Subtraction Subtracting a number is the same as adding the opposite of the number, i.e. b) (b = aa . This is why addition and subtraction have the same precedence in the order of operations. Subtraction is neither commutative nor associative, i.e. abba and c)(bacb)(a . If we view subtraction as addition by the opposite number, then we can move numbers around with the preceding operations. For example: dbcd = acba . Also you can group the subtraction part by using ba (a + b) = and ba b) = (a .

Example 1: ?14122610178132

Solu on: 252501210171314268214122610178132

Example 2: ?6043052027013020 =......

Solu on: ............ 60270130430520206043052027013020 15.005.020195.02.0604095.02.0 ...

Example 3: ?433823191411751

Solu on: 3843192311145714338231914117511554321

Mental Skill: Add or Subtract 99, 98, 999, 998, 997, 2995 etc. Adding or subtracting a number close to a good number is easy. For example:

52235253400125397125 126512152003215200321195321

352734572000234572000234519932345

Example 4: ?5997499639882975

Solu on: 36000450001240002530005997499639882975 -

1795644180003412256000500040003000

Example 5: ?1998499559963012

Solu on: 22000550004600012300019984995599630121325160254122000600050003000

Example 6: A copy of the MathTown Quarterly costs $1.99 more than a copy of the MathTown Monthly. Mr. Galois buys both magazines for $17.89. How much does the MathTown Monthly cost?

Chapter 1: Number Sense and Speed Calcula on --- Mental Skills for Problem Solving

8

Example 2: Joey plans to resume his summer job at PizzaWorld, which requires the use of his personal vehicle to deliver pizzas. Including tips, Joey earns an average of $18 per hour, and he anticipates spending $395 each month on gas and vehicle maintenance. If he will work for three months, what is the minimum number of hours he must work to earn $9443 for tuition and cover his gas and vehicle maintenance expenses?

Solu on on car: 11853953 .

He must make 1062811859443 dollars. The number of hours he must work is:

4.59018

10628 Round up the answer to 591.

Answer: 591 hours

Use mental technique for calculations: 1185151200540033953 ,

9490600

986600

9865400

95314

1810628

94590

9410600

94

990600

Round up to next whole number 591.

Mental Skills for Problem Solving Many of the word problems can be solved with some mental techniques. Here we will just go through some typical examples.

Example 1: There are some frogs and some lily pads at a pond. If the lily pads with frogs on them have four frogs each, then there are three lily pads with no frogs on them. If each lily pad has exactly three frogs on it, then there are four frogs with no lily pad. How many frogs are at the pond?

Solu on: Suppose that you are the Frog Master. Now the frogs are on lily pads with four frogs on each lily pad but 3 of the lily pads are empty. You command that one frog from each lily pad

jump up to your shoulder. If you put 3 frogs from your shoulder on each of the 3 empty lily pads you will still have 4 left on your shoulder. So you have 3x3 + 4 = 13 frogs jumped to your shoulder. 4x13 = 52 is the number of frogs at the pond. Answer: 52

Example 2: The heights of the five starters of a high school basketball team are 6'4", 6 5 , 6 8 , 6 11 and 7 . What is the average height of these players, in inches?

Solu on: Do not hasten to add the heights then divide the total by 5. Think of the 7 height as 6 12 so you just need to figure out the average of the inches. Still do not add yet. Noting that

the average of 4 is 8 and the average of 5 and 11 is also 8 you will immediately see the average of the 5 heights is 6 8 Convert to inches: 6 8 = 6x12 + 8 = 80 inches. Answer: 80

Example 3: There are a total of 81 men and women in a store. There are 5 more men than women. How many women are in the store?

Solu on: Remove 5 men from the store so you will end up with equal number of men and women in the store. 81 5 = 76 total there. Divide 76 by 2 and you get 38. Answer: 38

Note: If you are asked to find the number of men, you want to add 5 extra women to the store!

Chapter 8: A Li le Bit of Analy c Geometry --- Number Lines and the Coordinate System

207

Example 10 What is the area, in square units, of triangle ABC pictured on the right? Express your answer as a common fraction.

Solu on

As shown in the figure on left, ABC is inscribed in a rectangle with sides parallel to the two axes. The area of ABC is the area of the rectangle (actually it is a square here) minus the areas of the three shaded right triangles.

.

Answer:

Actually there is a faster way to find the answer for this problem. There is a formula for finding the area of any polygon in a coordinate system if the coordinates of the vertices are given.

Shoelace Theorem: If the vertices of a polygon in either clockwise or counterclockwise order on the polygon are 111 , yxP , 222 , yxP , 333 , yxP , , nnn yxP , then the area of the polygon is given by:

nnnnnn yxyxyxyxyxyxyxyxA 11231211322121

This formula looks complicated and useless unless we know why people call it the Shoelace Formula or Shoestring Formula.

Shoelace Algorithm Step 1: Stack the n points vertically in order and add the first point at the bottom. Step 2: Multiply the coordinates in a shoelace way shown on right, i.e. multiply diagonally down and to the right; multiply diagonally down and to the left Step 3: Add the products on left side to obtain the sum L; add the products on right side to obtain the sum R. Step 4: Half of the absolute difference between L and R is the area of the polygon.

213

2382

2361641

2131

2143

2144

213


208

Applying this shoelace method to the triangle in previous example we should get something below:

Example 11: What is the area of the quadrilateral below?

Solu on:

The Shoelace Theorem is very convenient to use to find the area of any polygon but there is a simpler method if we have all the grid points shown and the first.

Example 12: What is the area of the following shaded polygon region?

Solu on: As shown on right, the polygon region can be divided into one rectangle and 5 right triangles. The area of the rectangle is 6. The area of the right triangle A is

2331

21

. The area of triangle B

is also 23

. The area of triangle C is

2. The area of triangle D is also 2. The area of triangle E is 1. The total is 14. Answer: 14


209

It will be nice to know that there is a theorem called Pick's Theorem, which states that on a grid with

points 1 unit apart, the area of a polygon on the grid is 12ba , where a is the number of points in

the interior and b is the number of points on the boundary. In our example, 10a and 10b , so the

area is 14151012

1010 . Note that the vertices of a polygon must be grid points in order for

Example 13: In the figure shown on right, the distance between adjacent dots in each row and in each column is 1 cm. In square centimeters, what is the area of the shaded region? Solu on: There are 9 dots inside the polygon, and there are 8 dots on

, the area of the shaded region is 9 + 8/2 - 1 = 9 + 4 - 1 = 12. Answer: 12 It is also as easy to find

Example 14: What is the area of the shaded letter A, as figured on right? Solu on use region plus the center triangle first: 12 + 8/2 1 = 15. The area of the center triangle is 2. The area of the shaded letter A is 13. Or we can find the total area of the shaded letter A together with the center triangle and the bottom trapezoid. The total area is (6x7)/2 = 21. The center triangle has area 2, bottom trapezoid has area 6. The area we are looking for is 21 2 6 = 13.

Example 15: An equilateral triangle has two vertices at (0, 3) and (6, 3). If the third vertex is in the first quadrant, what is its y-coordinate? Express your answer in simplest radical form.

Solu on: Remember the special

right triangle: side ratio: . Also remember the height of the equilateral

triangle is . The height of our equilateral

triangle is 33623

.

Because the third vertex is in the first quadrant it will be above the line segment drawn. The y-coordinate of the third vertex is 333 . Answer: 333

9060302:3:1

a23

Chapter 10: Prac ce Tests --- Prac ce Test #5

258

19. _________

One line has a slope of 21

and contains the point (3, 9). Another line has a

slope of 52

and contains the point (1, 1). What is the sum of the coordinates

of the point at which the two lines intersect?

20. _________ m and n are positive integers such that 572am and 322an for some real number a . What is the value of the product mn?

21. _________ A room of 10 feet long and 5 feet wide has a height of 5 feet. An ant wants to

go from a corner A on the floor to the farthest corner at the ceiling B. What is the length of the shortest path from A and B, in feet? Express your answer in simplest radical form.

22. _________ A standard die is rolled three times with respective outcomes a , b and c .

What is the probability that cba ? Express your answer as a common fraction.

23. _________ An arithmetic sequence of positive integers has nineteen terms and a sum of

2014. What is the maximum possible value of any number of the sequence?

24. _________ If 2ab , 3bc and 4ac , what is the maximum possible value cba 23 ?

Express your answer in simplest radical form.

Chapter 10: Prac ce Tests --- Prac ce Test #5

259

25. _________ If x and y satisfy the equation 50125 22 yx , what is the minimum

possible value of yx ?

26. _________ A sequence of 10 integers 1a , 2a , , 10a starts with 01a . For all 102 i ,

2ia and either 11ii aa or 11ii aa . How many such sequences

exist?

27. _________ From each side of the regular hexagon ABCDEF a

semicircle is drawn inside the hexagon with the side as the diameter, as shown. If the side length of the hexagon is 2, what is the area of the center shaded region? Express your answer in terms of

and in simplest radical form.

28. _________ How many whole numbers between 1 and 2014 have more even factors than

odd factors?

29. _________ A circle is inscribed in a rhombus with side length of 12 units. The two acute

angles of the rhombus measure 60 degrees. Two smaller circles are then inscribed in the figure as shown. What is the radius of the two smaller circles? Express your answer in simplest radical form.

30. _________ m is the sum of those positive integers n such that n1

has exactly 13 digits

after the decimal point. What is the units digit of m ?

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