Your total score is based on the three problems for which you earn the most points; the scores for the individual parts of a single problem are summed. Points for each problem are shown in brackets [ ].
1. [3] Find all the real roots of the equation (x+1)21 +(x+1)20 (x-1)+(x+1)19 (x-1)2 + ... +(x-1)21 = 0.
2. [3] The bases of a trapezoid have integer lengths. Prove that one can cut the trapezoid into equal triangles.
3. [6] A circular disk and a point A within it are given. Find the locus of vertices C of all possible rectangles ABCD, where B and D lie on the boundary of the disk.
4. [7] Two highway robbers called Grabber and Eyer are dividing their loot, which is a heap of 100 coins. Grabber grabs a handful of coins from the heap, then Eyer decides who gets this handful. This goes on until one of them gets 9 handfuls. After that the other one gets all the remaining coins (it may happen that every- thing is divided up before anyone gets the 9 handfuls). Each time, Grabber can grab as many coins as he wishes. What is the largest number of coins that he can be sure of getting no matter what Eyer does? (Give this number, explain what exactly should Grabber do to make sure that get this number, and prove that he cannot be sure to get more.)
5. [7] What is the maximal number of knights that one can put on a 5×5 chess board so that each one attacks exactly two other ones? (Give such a position, and prove that one cannot find a position with a greater number of knights without violating the conditions of the problem.)
6. [10] In a chess tournament each participant plays with each of the other ones exactly once. A win is worth one point, a draw is worth half a point, a loss, zero points. Let us call a game an upset if the player who won has gained less points in the total score than the player who lost. Prove that no matter what the results of the tournament, strictly less than 3/4 of games were upsets.
1. [3] The integers n, m
are coprime (i.e., they have no common divisor greater than one).
The fraction
(m + 2000n)/(n + 2000m)
can be simplified by cancelling the common divisor d.
What is the maximal possible value of d?
2. [5] The chords AC and BD of a circle with center O intersect at the point K. The points M and N are the centers of circles circumscribed around triangles AKB and CKD. Prove that OM = KN.
3. [5] In a pack of cards, some are face-up, while the others are face-down. From time to time Peter takes a stack of several consecutive cards from the pack, so that the top and the bottom cards of the pack are face-up. Then he turns the whole stack around and puts it back into the pack exactly in the same place. Prove that eventually all cards will be lying face-down. (Remark: if a "stack" consists of only one single card, the only requirement is that it was face-up.)
4. [5] The plane is split into square cells by vertical and horizontal lines. A convex polygon is drawn on the plane in such a way that all its vertices are at vertices of some cells, and all the sides are neither horizontal nor vertical. Prove that the sum of the lengths of the pieces of vertical lines that lie within the polygon is equal to the sum of the lengths of the pieces of horizontal lines that lie within the polygon.
5. [7] Find the maximal number N for which there exist N consecutive positive integers such that the sum of digits in the first integer is divisible by 1, the sum of digits in the second integer is divisible by 2, the sum of digits in the third integer is divisible by 3, and so on, the sum of digits of the Nth integers is divisible by N.
6. In a chess tournament each participant
plays with each of the other ones exactly once. A win is worth
one point, a draw is worth half a point, a loss, zero points. Let
us call a game an upset if the player who won has gained less
points in the total score than the player who lost.
a) [6] Prove that no matter what the results of
the tournament, strictly less than 3/4 of
games were upsets.
b) [6] Prove that one cannot replace the number 3/4
in (a) by a smaller number.