AwesomeMath Admission Test Cover Sheet Your Name Last Name Admission Test Contact Information Please Print First Name A Phone Number Email Number of pages (not including this cover sheet) B C Check one AwesomeMath Test A January 17 - February 7, 2014 • Do not be discouraged if you cannot solve all of the questions: the test is not made to be easy. We want to see the solutions you come up with no matter how many problems you solve. • Include all significant steps in your reasoning and computation. We are interested in your ability to present your work, so unsupported answers will receive much less credit than well-reasoned progress towards a solution without a correct answer. • In this document, you will find a cover sheet and an answer sheet. Print out each one and make several copies of the blank answer sheet. Fill out the top of each answer sheet as you go, and then fill out the cover sheet when you are finished. Start each problem on a new answer sheet. • All the work you present must be your own. • Do not be intimidated! Some of the problems involve complex mathematical ideas, but all can be solved using only elementary techniques, admittedly combined in clever ways. • Be patient and persistent. Learning comes more from struggling with problems than from solving them. Problem-solving becomes easier with experience. Success is not a function of cleverness alone. • Submit your solutions by e-mail (preferred) by Friday, February 7, 2014 or by mail so that they arrive in our office by February 7, 2014. • Make sure that the cover sheet is the first page of your submission, and that it is completely filled out. Solutions are to be mailed to the following address: Dr. Titu Andreescu 3425 Neiman Road Plano TX 75025 If you e-mail your solutions, please send them to tandreescu@gmail.com E-mailed solutions may be written and scanned or typed in TeX. They should be sent as an attachment in either .doc or .pdf format. If you write and scan your solutions, insert the scans into a .doc or .pdf file and send just the one file. Please go to the next page for the problems Test A January 17 — February 7 1. Prove that 2014 can be written as (a2 + b2 )(c3 − d3 ) for some positive integers a, b, c, d. 2. Find all four-digit numbers n whose sum of digits is equal to 2014 − n. 3. Is there a perfect cube that can be represented as n + 2014 , for some positive integer n? n 4. Given an 11-letter “word”, a computer is programmed to randomly remove 7 of its letters. What is the probability that, if the input is AWESOMEMATH, the output is WEST? 5. Let a, b, c be positive real numbers such that 1 1 1 2014 + + = . a b c a+b+c Evaluate � a� � b� � c� 1+ 1+ 1+ . b c a 6. Find the largest possible power of 2 that could divide the number 99 . . . 904, containing n 9’s. 7. Let a, b, c, d, e be an arithmetic sequence such that a−b+c+d+e = 2014 and a2 +b2 +c2 = d2 +e2 . Find a. 8. Find all pairs (m, n) of integers such that m2 + mn + n2 = 13. 9. Solve in real numbers the equation x3 + �x�3 + {x}3 = 6x�x�{x}, where �x� and {x} denote the greatest integer less than or equal to x and the fractional part of x, respectively. 10. Let ABCD be a convex quadrilateral inscribed in a semicircle of diameter AD = 2. Prove that AB 2 + BC 2 + CD2 + AB · BC · CD = 4. AwesomeMath Answer Sheet Your Name Problem Number Page Of Write neatly! All work should be inside the box. Do NOT write on the back of the page!