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 2012-gon *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

Fred Flintstone and Barney Rubble in turn add pebbles to a pile. On each turn, they must add at least one pebble and may not add more pebbles than there are already in the pile. The player who makes the pile consist of exactly 2007 pebbles wins. Find a strategy that allows Fred or Barney to win regardless of how the other may play. Oh, there is originally just one pebble in the pile, and Fred goes first.

2. Secrets of Tables

Each cell of an 8 x 8 chessboard is filled with a 0 or 1. Prove that if we compute sums of numbers in each row, each column, and each of the two diagonals, we will get at least three equal sums.

3. More Secrets of Tables

(a). Prove that no matter how each cell of a 5 x 41 table is filled with a 0 or 1, one can choose 3 rows and 3 columns which intersect in 9 cells filled with identical numbers.

(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

A number is placed in each angle of a regular 2007-gon so that the sum of any 10 consecutive numbers is positive. Prove that one can choose an angle with the number *a* in it, such that when we label all 2007 numbers clockwise *a=a1, a2, …, a2007,* each sum *a1, a1 + a2, …, a1+ a2+ … +a2007* will be positive.

5. Natural Split

Prove that if *a* and *b* are positive irrational numbers with 1/*a* + 1/*b* = 1, then the two sets *A =* {[1*a*], [2*a*],…[*na*], …} split the set *N* of positive integers.

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.*

1. On a Collision Course

93 identical balls move along a line, 59 of them from left to right with speed v; the remaining 34 balls move from right to left toward the first group of balls with speed w. When two balls collide, they exchange their speeds and direction of motion. What is the total number of collisions that will occur?

2. A Horse! A Horse! My Kingdom for a Horse!

The Good, the Bad, and the Ugly divide a pile of gold and a horse. The pile consists of 2006 gold coins, and they draw in turn 1, 2, or 3 coins from the pile. The Good gets the first turn, the Bad draws second, and the Ugly takes last. The Ugly does not trust the Bad and never draws the same number of coins as the Bad has drawn immediately before him. The one who takes the last coin is left behind, while the two others cross the prairie together on horseback. Who can guarantee himself the ride out on the horse regardless of how the others draw the coins? How can he do this?

3. Chess Tournament

Professional and amateur players, n in all, participated in a round robin chess tournament. Upon its completion, it was observed that each player earned half of his total score in games against the amateurs. Prove that Ön is an integer.

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

The Denver Mint manufactured 2006 silver dollars. As a quality control procedure, every pair of consecutively made coins was weighed against each other, and the difference in weight was less than 1/100 of an ounce. Prove that the coins can be partitioned in such a way that the total group weights differ by less than 1/100 of an ounce.

5. Math Party

We say that mathematicians a and b have each other’s number n if there is a coauthor-chain of mathematicians a = a0,a1,…,an = b such that every consecutive pair in the chain are coauthors on at least one publication.

A party is attended by at least three mathematicians, and every pair of attendees has a coauthor-chain at the party. Prove that all mathematicians can be seated at a round table in such a way that every pair of neighbors has each other’s number not exceeding 3.

Copyright ® 2006, by the Center for Excellence in Mathematical Education, 885 Red Mesa Drive, Colorado Springs, CO 80906, USA

1. Cover-up

Can a square of area 2005 be covered bt 401 squares of area 5 each?

2. Tea Time

Is it possible to arrange the numbers 1 to 100 in a 10X10 grid so that the entries of any T-shaped figure consisting of 4 unit squares of the grid sum up to an even number?

3. One L of a Grid

What is the minimum number of squares to be colored red in a 10X10 grid so that any L-shaped figure consisting of 4 unit squares of the grid contains at least 2 red squares?

4. Black and Blue

(a) Each vertex of a regular 11-gon is colored black or blue. Prove that there are two congruent monochromatic triangles of athe same color.

(b) Each vertex of a regular 2005-gon is colored red or blue. Prove that there are two congruent monchromatic 10-gons 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

(a) The DNA of bacterium bacillus anthracis(causing anthrax) is a sequence, each term of which is one of 2005 genes. How long can the DNA be if no consecutive terms may be the same gene, and no two distinct genes can reappear in the same order? That is, if distinct genes *a*, *ß* occur in that order (with or with out any number of genes in between), the order *a*, *…, ß* cannot occur again.

(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

1. Homeland Security

Intelligence reports that terrorists have sailed a 1 x 5 frigate completely within our coastline which is a 2004-long strip of 1 x 1 cells. Naturally, intelligence does not know exactly where the frigate is. What is the smallest number of guided missiles we must launch to destroy the enemy’s frigate? (A missile destroys a 1 x 1 cell, and a single hit on the frigate destroys it.)

2. 2004 Tails

You are in an absolutely dark room with two tables, A and B. On top of A there is an unknown huge number of silver dollars, exactly 2004 of which lie tails up. Nothing is on top of B. You may conduct two operations: you may move any number of dollars from one table to another; you may flip over any number of dollars on one of the tables. (You cannot tell in any way whether a coin is tails or heads up.)

(a) Can you obtain an equal number of tails on the tables through a series of allowed operations?

(b) Can you obtain an equal number of heads on the tables through a series of allowed operations?

3. Crawford Cowboy Had a Farm, Ee-i-ee-i-o

A Crawford cowboy is unable to create a rectangular ranch using all 2004 straight sections of fence that he has. He decides to break some of the sections, each section no more than once, so that the resulting set can do the job.

(a) Prove that there is a collection of 2004 sections such that even after breaking exactly one section, he cannot create a rectangular ranch using all the sections.

(b) Find the smallest number n, such that breaking a sections is enough to do the job regardless of the lengths of the original 2004 sections.

4. To Have a Cake

(a) We need to protect from the rain a cake that is in the shape of an equilateral triangle of side 2.1. All we have are identical tiles in the shape of an equilateral triangle of side 1. Find the smallest number of tiles needed.

(b) Suppose the cake is in the shape of an equilateral triangle of side 3.1. Will 11 tiles be enough to protect it from the rain?

5. Chess 7X7

(a) Each member of two 7-member chess teams is to play once against each member of the opposing team. Prove that as soon as 22 games have been played, we can choose 4 players and scat them at a round table so that each pair of neighbors has already played.

(b) Prove that 22 is the best possible; i.e., after 21 games the result of (a) cannot be guaranteed.

Copyright ® 2004, by the Center for Excellence in Mathematical Education, 885 Red Mesa Drive, Colorado Springs, CO 80906, USA