How to Excel at Middle School Math Competitions Huasong Yin

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1 MathCounts Preparation How to Excel at Middle School Math Competitions By Huasong Yin

2 0, 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-: ISBN-0: Printed in the United States Print Date: /6/0 for any errors or mistakes

3 Table of Contents Chapter Number Sense and Speed Calculation... Addition... Subtraction... Multiplication... Division... 7 Mental Skills for Problem Solving... 8 Translating from Verbal to Algebra... Chapter Exponents and Fractions... 7 Exponential Notation and Properties of Exponents... 7 Place Values and Number Systems... Powers of Ten and Scientific Notation... Divisibility, Prime Numbers and Prime Factorization... 8 Fractions... Ratio, Proportion and Percent... Speed - Distance Time Chapter Probability, Counting and Basic Statistics... 6 Sets, Subsets, Set Operations and Logic... 6 Counting Techniques, Permutations and Combinations... 7 Probability... 9 Mean, Median, Mode, Range and Other Means... 0 Chapter Elementary Algebra... 0 Variables and Algebraic Expressions... 0 Using (a + b) = a + ab + b... Difference of Two Squares: Using (a + b)(a b) = a b... 6 Square Roots and Simplifying Square Root Expressions... 9 Chapter Geometry... Perimeter and Area... Angles, Degree Measurement and Polygons... 7 Triangles... Pythagorean Theorem and Special Right Triangles... 9

4 Chapter : Number Sense and Speed Calcula on --- Addi on Triangles and Trapezoids... 9 Circles... Solid Objects, Polyhedrons, Volumes and Euler Theorem... 9 Chapter 6 Sequences... 6 Sequences and Pattern Recognition... 6 The Story of Little Gauss and Arithmetic Sequences... 6 The Smart Pizza Eater and Geometric Sequences Other Sequences... 7 Chapter 7 Equations, Inequalities and Functions Linear Equations Quadratic Equations Inequalities... 9 Functions Chapter 8 A Little Bit of Analytic Geometry... 0 Number Lines and the Coordinate System... 0 Lines and Circles in a Coordinate System... Chapter 9 A Little Bit of Number theory... 6 Repeating Decimals... 6 Modular Arithmetic... 0 Chapter 0 Practice Tests... Some Test-Taking Tips... Practice Test #... Practice Test #... 0 Practice Test #... Practice Test #... 0 Practice Test #... Solutions to Practice Test Solutions to Practice Test... 6 Solutions to Practice Test Solutions to Practice Test Solutions to Practice Test... 8

5 Chapter : Number Sense and Speed Calcula on --- Addi on Chapter Number Sense and Speed Calculation In the MathCounts Sprint round you are given 0 questions to finish in 0 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 a + b = b + a and a + b + c = a + b + c for any numbers a, b and c. For example: + = +, = 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 : =? Solu on: = = = 0 Example : =? Solu on: = ( )+ ( ) + (.86 +.) +. = +++.=+. =. Example :? Solu on: 0 Example : =? Solu on: = (+ 9) + ( +8) +(+ 7) + ( + 6) + = = Example : =? Solu on: =

6 Chapter : Number Sense and Speed Calcula on --- Subtrac on Subtraction Subtracting a number is the same as adding the opposite of the number, i.e. a b = a ( b). This is why addition and subtraction have the same precedence in the order of operations. Subtraction is neither commutative nor associative, i.e. a b b a and (a b) c a (b c). If we view subtraction as addition by the opposite number, then we can move numbers around with the preceding operations. For example: a b c d = a c b d. Also you can group the subtraction part by using (a + b) = a b and (a b) = a b. Example : ? Solu on: Example : =? Solu on: Example : 7 9 8? Solu on: Mental Skill: Add or Subtract 99, 98, 999, 998, 997, 99 etc. Adding or subtracting a number close to a good number is easy. For example: Example : ? Solu on: Example : ? Solu on: Example 6: A copy of the MathTown Quarterly costs $.99 more than a copy of the MathTown Monthly. Mr. Galois buys both magazines for $7.89. How much does the MathTown Monthly cost?

7 Chapter : Number Sense and Speed Calcula on --- Mental Skills for Problem Solving Example : 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 $8 per hour, and he anticipates spending $9 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 $9 for tuition and cover his gas and vehicle maintenance expenses? Solu on on car: 9 8. He must make dollars. The number of hours he must work is: Round up the answer to 9. 8 Answer: 9 hours Use mental technique for calculations: , Round up to next whole number 9. 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 : 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 of the lily pads are empty. You command that one frog from each lily pad jump up to your shoulder. If you put frogs from your shoulder on each of the empty lily pads you will still have left on your shoulder. So you have x + = frogs jumped to your shoulder. x = is the number of frogs at the pond. Answer: Example : The heights of the five starters of a high school basketball team are 6'", 6, 6 8, 6 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. Think of the 7 height as 6 so you just need to figure out the average of the inches. Still do not add yet. Noting that the average of is 8 and the average of and is also 8 you will immediately see the average of the heights is 6 8 Convert to inches: 6 8 = 6x + 8 = 80 inches. Answer: 80 Example : There are a total of 8 men and women in a store. There are more men than women. How many women are in the store? Solu on: Remove men from the store so you will end up with equal number of men and women in the store. 8 = 76 total there. Divide 76 by and you get 8. Answer: 8 Note: If you are asked to find the number of men, you want to add extra women to the store! 8

$Express your answer as a common fraction. Solu on As shown in the figure on left, ABC is inscribed in a rectangle with sides parallel to the two axes.$ 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.

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.

Shoelace Theorem: If the vertices of a polygon in either clockwise or counterclockwise order on the polygon are P x, y, P x, y, P x, y,, P n xn, yn then the area of the polygon is given by: A xy x y

8 Chapter 8: A Li le Bit of Analy c Geometry --- Number Lines and the Coordinate System Example 0 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 P x, y, P x, y, P x, y,, P n xn, yn then the area of the polygon is given by: A xy x y xn yn xn y x y xy xn yn x This formula looks complicated and useless unless we know why people call it the Shoelace Formula or Shoestring Formula. Shoelace Algorithm Step : Stack the n points vertically in order and add the first point at the bottom. Step : 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 : Add the products on left side to obtain the sum L; add the products on right side to obtain the sum R. Step : Half of the absolute difference between L and R is the area of the polygon. 6 6 y n 8. 07

Chapter 8: A Li le Bit of Analy c Geometry --- Number Lines and the Coordinate System Applying this shoelace method to the triangle in previous example we should get something below: Example :

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.

9 Chapter 8: A Li le Bit of Analy c Geometry --- Number Lines and the Coordinate System Applying this shoelace method to the triangle in previous example we should get something below: Example : 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 : 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 right triangles. The area of the rectangle is 6. The area of the right triangle A is. The area of triangle B is also. The area of triangle C is. The area of triangle D is also. The area of triangle E is. The total is. Answer: 08

Chapter 8: A Li le Bit of Analy c Geometry --- Number Lines and the Coordinate System It will be nice to know that there is a theorem called Pick's Theorem, which states that on a grid with b points

Note that the vertices of a polygon must be grid points in order for Example : In the figure shown on right, the distance between adjacent dots in each row and in each column is cm.

10 Chapter 8: A Li le Bit of Analy c Geometry --- Number Lines and the Coordinate System It will be nice to know that there is a theorem called Pick's Theorem, which states that on a grid with b points unit apart, the area of a polygon on the grid is a, where a is the number of points in the interior and b is the number of points on the boundary. In our example, a 0 andb 0, so the 0 area is 0 0. Note that the vertices of a polygon must be grid points in order for Example : In the figure shown on right, the distance between adjacent dots in each row and in each column is 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/ - = =. Answer: It is also as easy to find Example : What is the area of the shaded letter A, as figured on right? Solu on use region plus the center triangle first: + 8/ =. The area of the center triangle is. The area of the shaded letter A is. 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)/ =. The center triangle has area, bottom trapezoid has area 6. The area we are looking for is 6 =. Example : An equilateral triangle has two vertices at (0, ) and (6, ). 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 a. The height of our equilateral triangle is 6. 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. Answer: 09

11 Chapter 0: Prac ce Tests --- Prac ce Test # 9. One line has a slope of and contains the point (, 9). Another line has a slope of and contains the point (, ). What is the sum of the coordinates of the point at which the two lines intersect? 0. m and n are positive integers such that m a 7 and n a for some real number a. What is the value of the product mn?. A room of 0 feet long and feet wide has a height of 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.. A standard die is rolled three times with respective outcomes a, b and c. What is the probability that a b c? Express your answer as a common fraction.. An arithmetic sequence of positive integers has nineteen terms and a sum of 0. What is the maximum possible value of any number of the sequence?. If ab, bc and ac, what is the maximum possible value a b c? Express your answer in simplest radical form. 8

Chapter 0: Prac ce Tests --- Prac ce Test #. If x and y satisfy the equation x y 0, what is the minimum possible value of x y? 6. A sequence of 0 integers a, a,, a 0 starts with a 0.

12 Chapter 0: Prac ce Tests --- Prac ce Test #. If x and y satisfy the equation x y 0, what is the minimum possible value of x y? 6. A sequence of 0 integers a, a,, a 0 starts with a 0. For all i 0, a and either a or a. How many such sequences i exist? i a i i a i 7. 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, what is the area of the center shaded region? Express your answer in terms of and in simplest radical form. 8. How many whole numbers between and 0 have more even factors than odd factors? 9. A circle is inscribed in a rhombus with side length of 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. 0. m is the sum of those positive integers n such that n has exactly digits after the decimal point. What is the units digit of m? 9

Moore Catholic High School Math Department

Moore Catholic High School Math Department Geometry Vocabulary The following is a list of terms and properties which are necessary for success in a Geometry class. You will be tested on these terms during