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 Awesome Math Test A March 16 - April 6, 2012 • 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. • Make sure that the cover sheet is the first page of your submission, and that it is completely filled out. Solutions are to be emailed to tandreescu@gmail.com or mailed to the following address: Dr. Titu Andreescu 3425 Neiman Road, Plano TX 75025 E-mailed solutions may be written and scanned or typed in LaTeX. 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 16 — April 6 1. The sum of some consecutive integers is 2012. Find the smallest of these integers. 2. How many 5-digit numbers have at least one digit of 5? 3. Find all integers n for which n2 − n + 1 divides n2012 + n + 2001. 4. Let a and b be real numbers such that 2a2 + 3ab + 2b2 ≤ 7. Prove that max(2a + b, a + 2b) ≤ 4. 5. Find all pairs (m, n) of integers such that m3 + n3 = 2015. 6. Let P (x) = 3x3 − 9x2 + 9x. Prove that P (a2 + b2 + c2 ) ≥ P (ab + bc + ca) for all real numbers a, b, c. 7. For a positive integer N , let r(N ) be the number obtained by reversing the digits of N . For example, r(2013) = 3102. Find all 3-digit numbers N for which r2 (N ) − N 2 is the cube of a positive integer. 8. Solve the system of equations 8 9 5 , log yz = , log zx = . log xy = log z log x log y 9. In circle C chords AB and XY intersect at P . Prove that the projections of P onto AX, BX, AY , BY are concyclic if and only if the midpoints of AX, BX, AY , BY are the vertices of a rectangle. 10. (a) Give example of a triple (a, b, c) of even positive integers such that ab + 1, bc + 1, ca + 1 are all perfect squares. (b) Are there triples (a, b, c) of odd positive integers such that ab + 1, bc + 1, ca + 1 are all perfect squares? 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!