COPYRIGHT Brandon Wang. No distribution other than through BrWang.com shall be allowed. PROBLEMS GUIDE Mathcounts / Contest Math This material was created and copyrighted by Brandon Wang. No distribution other than through BrWang.com shall be allowed. You are permitted to download and print this material for study usage only. 2011 Brandon Wang. All rights reserved. RESOURCE TIMESTAMP: This document was generated on 2/5/2011. DISCLAIMER: This resource is made without any warranty or guarantee of accuracy. Information on this resource This is a problem guide that goes over some types of contest math problems found in the Mathcounts program. These most likely appear at School, Chapter, and State levels of the program. You might also find them in the General Math portion of the TMSCA program. This is by no means complete by any means. Study guide The problems guide begins on the next page. 1
Course Summaries by Brandon Wang [brwang.com] Page 1 Mathcounts Program / Contest Math Basics o o, NOT On scratch paper, write top to bottom, left to right. Mark problems clearly. Try not to write work where your hand will obstruct it o Always write out the original equation, with originally known numbers Don t simplify or calculate. Use original information After you write it out, you can simplify QUICKLY o The purpose is to facilitate checking easily Write what it is (ex. ) o A big complex problem? Try to write a simpler dumbed-down version of the problem For example, if need to find 2 3 4 6 o Try 2 first, then 2 3. Find a pattern o Tricky: In this specific one, you can factor 4 2 2! Look for a pattern through this Integer problems o Two digit number is equal to 10 (for 2-digit, range of is 1-9, range of is 0-9) Change the equation so there are no fractions or decimals In the equation, the units digits must be equal 10 s unit digit must be zero For example, 10 60 30 3 o The units digit of and 3 must be equal Any carry to tens digit is ignored o Two whole numbers add/multiply to a whole number. Find specific relationships, instead of combining to one large equation. Ex. (both sides are INTEGERS) o So, must be divisible by 12, which is 12 ( 1,2,3 ) o Similarly, the other side must be an integer:, so must be divisible by 5, which is 5 1,2,3 o Methods of guess-and-check Assume is 1. Try to find values for x and y. If they fail, assume is 2, and so on.
Course Summaries by Brandon Wang [brwang.com] Page 2 It is easier to try 12 because you can just find prime factors of 12 = 1 2 2 3, running through combos. Series of functions/unknowns, with given relationship between two points o Eg. 12 This function tells us three things: Known 1 and 2, you can calculate. o With known two consecutive, unlimited upwards calculations Known 2 1 o Actually is 21 o With known two consecutive, unlimited downward calculations Known 1 2 o Actually is 11 o With known two even/odd consecutive, you know the inbetween number, hence you know EVERYTHING Series of numbers o Generally: find relationship between the numbers as a general rule will help Subtract one point from the next one over to find the difference Write out enough samples to identify a pattern If that is not possible, try to find a common difference Re-plug into original list to test if your equation is correct o Important points 1 Eg. in the sequence,,,,, and 1, and each of the following terms is the sum of all previous terms of the sequence. What is the value of when 20? Each term can be named as or, or ANYTHING. o Asking for sum Eg. 13579 43454749 Each number (ignore negatives) is a arithmetic sequence Method number one o 1 and 49, -3 and -47, 5 and 45, come out to 50, -50, 50 so on. o Consecutive numbers add to, what are the numbers Whenever dealing with consecutive numbers, try to cancel out the numbers Eg. 7 consecutive positive numbers add to a perfect cube Dead end method: 1 2 6 o Equals 7 21 (the 21 is a problem) Instead, you should: 3 2 1 1 2 3 o Equals 7 (the numbers cancel out)
Course Summaries by Brandon Wang [brwang.com] Page 3 o So this is a perfect cube, and the 7 is a prime factor. Start trying 14, 21, 28, Chasing problems o Must start at different points, must have different speed, faster behind slower, where/when do they meet? One is behind the other, running in the same direction o time until meet o difference of starting points o speed of slower one, speed of faster one. Both at different places, running towards each other o Running in a circle / laps When will the meet? Assume at the beginning one is already ahead by one lap Using ( can be one lap, or 100m, etc.) When will they meet the second time? Using 2 (the other is now ahead by 2 laps) o Originally comes from 1, where n is the number of laps amount of laps the slower one runs 1 amount of laps the faster guy runs We drop the full laps to just get the position where they meet Intersections on Coordinate Plane o Two functions where / how many times do they meet? First, recognize the shape the functions are in (line, parabola, squiggle, etc.) Obtain equations for each form, and set them equal to. & Solve for possible x. o Every possible x is a solution count them up o Is zero a possibility? o For example, 0 o Factor out an x: 0 and factorize the part in parentheses o If factoring is hasslesome (more than 15 seconds) use the quadratic formula (only works for quadratic functions) 4 If negative no solutions for this portion. If zero one solution for this portion.
Course Summaries by Brandon Wang [brwang.com] Page 4 If positive two solutions for this portion. Are you counting the number of solutions? Don t forget to add on any other solutions (if applicable) Solving high-powered functions o Eg. 40 The goal is to LOWER the power of the function You can do this by factorizing for example Do not forget quadratic function factorization methods Understanding problems with absolute values o First step: determine whether the absolute value term is positive or negative o Remove the absolute values accordingly o if 0 o if 0 Multiplying digits together o Giving you four digits to multiply, what is the largest answer? Place the two largest as tens place Make the smaller number s one s place as LARGE as possible Geometry problems o Two points, must go back to x or y axis Reflect one point over the line and find the distance