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 B March 22 – April 12, 2013 • 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. • Postmark or submit your solutions by e-mail (preferred) by Friday, April 12, 2013. • 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 B March 22 — April 12 1. Let pn be the nth prime number, n = 1, 2, . . . Find the least even k for which p1 + p2 + . . . + pk is not a prime. 2. If the sum of the numbers 4a + 14 , 4b + 14 , 4c + 14 , 4d + 14 is 2013, what is the arithmetic mean of the numbers a + 14 , b + 14 , c + 14 , d + 14 ? 3. Find the least n such that 1 + 2 + . . . + n is divisible by 2013. 4. Is there a 14-digit perfect square of the form 2013abcdef ghij, where a, b, c, d, e, f , g, h, i, j are all distinct? 5. What is the greatest common divisor of the numbers 29 − 2, 39 − 3, . . ., 20139 − 2013? 6. Let n be an integer greater than 1. Simplify √ n3 + (n2 − 4) n2 − 1 − 3n2 + 4 √ . n3 + (n2 − 4) n2 − 1 + 3n2 − 4 � � 7. If nk denotes the number of combinations of n objects � �� taken k at a time, find the greatest 2-digit number n such that n3 n4 is a perfect square. 8. Solve the equation �x�2 + 4{x}2 = 4x − 5, where�x� and {x} are the greatest integer less than or equal to x and the fractional part of {x}, respectively. 9. Let a and b be non-negative real numbers. Prove that (a + b)5 ≥ 12ab(a3 + b3 ). 10. In triangle ABC, 2∠A = 3∠B. Prove that (a2 − b2 )(a2 + ac − b2 ) = b2 c2 . 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!