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 C May 2 – May 10 (UTD) May 2 – May 16, 2014 (Cornell & Berkeley) • 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 May 10, 2014 for UT Dallas applicants and May 16 for Cornell and Berkeley applicants or by mail so that they arrive in our office by the above mentioned deadlines • 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 [email protected] 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 C May 2 – May 10 (UTD) & May 2 – May 16 (Cornell, Berkeley) 1. Let a, b, c be integers such that the average of the numbers 7a − 9, 7b − 9, 7c − 9 is 2014. Prove that the average of the numbers a, b, c is a perfect square. 2. Find all primes p for which p6 + 6p − 4 is also prime. 3. How many three-digit numbers contain at least an odd digit? 4. Find all negative integers a for which the equation x2 + ax + 2014 = 0 has two integer roots. 5. Find the greatest primes p and q, p > q, such that p3 and q 3 divide 30! + 29! . 28 6. Find the number of positive integers n less than 100 for which (n + 1)2 divides (2n + 1)! 7. Find all integers n such that n − 2014 and n + 2014 are both triangular numbers. 8. Let a, b, c be real numbers such that a2 + b2 + c2 = 1. Find the maximum possible value (a + b)c. 9. Find the number of ordered pairs (m, n) of positive integers such that mn = 2010020020010002 (no calculators allowed). 10. Prove that 1 1 ◦ + sin 9 cos 9◦ √ √ can be written as m + n for some positive integers m and n. 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!