Sample Problems


April 20, 2012 These problems were selected and edited for you by the Problem Committee: Robert Ewell, Gary Miller, and Alexander Soifer. All problems were created especially for this Olympiad by Alexander Soifer. Present complete solutions. We will give no credit for answers submitted without supporting work. Conversely, a minor error that leads to an incorrect answer will not substantially reduce your credit. 1. Summing Up (A) Suppose three integers are added pairwise, and the results are 403, 704, and 905. Find the integers if they exist. (B) Solve the same problem if the pairwise sums are 403, 704, and 906. 2. Triangulation The convex 2012gon P is partitioned into finitely many triangles so that each side of P belongs to one triangle, and no vertex of any triangle lies in an inner point of a side of another triangle. Can the map thus formed be colored in two colors, red and blue, so that no triangles which share a side are the same color and all the triangles along the perimeter of P are blue? 3. Just Your Average Cube Each corner of the initial cube contains a number, and the numbers include 2012 and 2013. An averaging operation creates the second cube by replacing each number by the mean of the three numbers one edge away from it. We then repeat the averaging operation obtaining the third cube, etc. (A) Can it so happen that all the numbers of the 2012^{th} cube coincide with the corresponding numbers of the initial cube? If yes, find all possible arrays of the corner numbers. (B) Can it so happen that all the numbers of the 2013^{th} cube coincide with the corresponding numbers of the initial cube? If yes, find all possible arrays of the corner numbers. 4. Beyond the Finite I (A) Given 11 real numbers in the form of infinite decimal representations, prove that there are two of them which coincide in infinitely many decimal places. (B) Show that 11 is the lowest possible quantity to guarantee the result. 5. Beyond the Finite II Infinitely many circular disks of radius 1 are given inside a bounded figure in the plane. Prove that there is a circular disk of radius 0.9 that is contained in infinitely many of the given disks. A circular disk consists of all points on and inside of a circle. 
April 20, 2007 These problems were selected and edited for you by the Problem Committee: Rovert Ewell, Gary Miller and Alexander Soifer. Problems 1,2 and 4 were created by Alexander Soifer for this Olympiad. Problem 3 was adapted by Soifer from Russian mathematical folklore. Problem 5 was first published in 1894 in "The Theory of Sound" by the British physicist John William Strutt, Lord Rayleigh, who discovered argon and won the 1904 Nobel Prize for it. 1. Stone Age Entertainment Returns 2. Secrets of Tables 3. More Secrets of Tables (b). Prove that 41 in part (a) is the lowest possible; i.e., the statement is not true for a 5 x 40 table. 4. Looking for the Positive 5. Natural Split A number a is called irrational if it cannot be presented in a form a=p/q with p, q integers and q = 0. The symbol [c] stands for the largest whole number not exceeding c. We say that sets A and B split N if each positive integer n is in either A or B but not in both A and B. 
April 21, 2006 These problems were selected and edited for you by the Problem Committee: Robert Ewell and Alexander Soifer. Problem 1 was created by Prof. Gregory Galperin of Eastern Illinois University on the occasion of Prof. Yakov Sinai's 70th birthday. Problems 2 and 5 were created by Alexander Soifer for this Olympiad; in fact, problem 5 is a "translation" into a story of the 1960 theorem by Prof. M. Sekanina from Czechoslovakia. Problems 3 and 4 were adopted by Soifer from the Russian mathematical folklore (see, for example, the 1970 booklet "Mathematical Competitions" by E. B. Dynkin, et al.). Present complete solutions. We will give no credit for answers submitted without supporting work. Conversely, a minor error that leads to an incorrect answer will not substantially reduce your credit. 
1. On a Collision Course 
2. A Horse! A Horse! My Kingdom for a Horse! 
3. Chess Tournament Round robin is a tournament where every pair of participants plays each other once. A winner of a game earns 1 point; the loser 0; a draw gives 1/2 a point to each player. 
4. The Denver Mint 
5. Math Party 
Copyright ® 2006, by the Center for Excellence in Mathematical Education, 885 Red Mesa Drive, Colorado Springs, CO 80906, USA 
April 22, 2005 These problems were selected and edited for you by the Problem Committee: Robert Ewell, Gary Miller, and Alexander Soifer. Problem 2 was created for this Olympiad by Ming Song of Colorado Springs. Problems 1, 3, 4, and 5 were created by Alexander Soifer for this year's Olympiad (problem 3 was inspired by Martin Klazar's research talk Soifer heard in Prague in 1996). Present complete solutions. We will give no credit for answers submitted without supporting work. Conversely, a minor error that leads to an incorrect answer will not substantially reduce your credit. 
1. Coverup 
2. Tea Time 
3. One L of a Grid 
4. Black and Blue (b) Each vertex of a regular 2005gon is colored red or blue. Prove that there are two congruent monchromatic 10gons of the same color. A polygon is called a monochromatic if all of its vertices are colored in the same color. Distinct polygons may have some but not all vertices in common. 
5. Love and Death (b) The DNA of bacterium Bacillus amoris 9 (causing love) is a sequence, each term of which is one of 2005 genes. No three consecutive terms may include the same gene twice, and no three distinct genes can reappear in the same order. That is, if distinct genes a,ß, and y occur in that order (with or wihtout any number of genes in between), the order a, ..., ß, ..., y cannot occur again. Prove that this DNA is at most 12,032 long. 
Copyright ® 2005, by the Center for Excellence in Mathematical Education, 885 Red Mesa Drive, Colorado Springs, CO 80906, USA 