RU-ALLRUSSIAN 1972

ABCD is a rectangle. M is the midpoint of AD and N is the midpoint of BC. P is a point on the ray CD on the opposite side of D to C. The ray PM intersects AC at Q. Show that MN bisects the angle PNQ.

Given 50 segments on a line show that you can always find either 8 segments which are disjoint or 8 segments with a common point.

Find the largest integer n such that $4^{27} + 4^{1000} + 4^{n}$ is a square.

a, m, n are positive integers and a $>1$ . Show that if $\mathrm{a^{m} + 1}$ divides $\mathrm{a^{n} + 1}$ , then m divides n. The positive integer b is relatively prime to a, show that if $\mathrm{a^{m} + b^{m}}$ divides $\mathrm{a^{n} + b^{n}}$ then m divides n.

A sequence of finite sets of positive integers is defined as follows. $\mathrm{S_0 = \{m\}}$ , where $\mathrm{m > 1}$ . Then given $\mathrm{S_n}$ you derive $\mathrm{S_{n + 1}}$ by taking $\mathrm{k^2}$ and $\mathrm{k + 1}$ for each element k of $\mathrm{S_n}$ . For example, if $\mathrm{S_0 = \{5\}}$ , then $\mathrm{S_2 = \{7, 26, 36, 625\}}$ . Show that $\mathrm{S_n}$ always has $2^{\mathrm{n}}$ distinct elements.

Prove that a collection of squares with total area 1 can always be arranged inside a square of area 2 without overlapping.

O is the point of intersection of the diagonals of the convex quadrilateral ABCD. Prove that the line joining the centroids of ABO and CDO is perpendicular to the line joining the orthocenters of BCO and ADO.

9 lines each divide a square into two quadrilaterals with areas 2/5 and 3/5 that of the square. Show that 3 of the lines meet in a point.

A 7- gon is inscribed in a circle. The center of the circle lies inside the 7- gon. A, B, C are adjacent vertices of the 7- gon show that the sum of the angles at A, B, C is less than 450 degrees.

Two players play the following game. At each turn the first player chooses a decimal digit, then the second player substitutes it for one of the stars in the subtraction | **** - **** |. The first player tries to end up with the largest possible result, the second player tries to end up with the smallest possible result. Show that the first player can always play so that the result is at least 4000 and that the second player can always play so that the result is at most 4000.

For positive reals x, y let f(x, y) be the smallest of x, 1/y, y + 1/x. What is the maximum value of f(x, y)? What are the corresponding x, y?

P is a convex polygon and X is an interior point such that for every pair of vertices A, B, the triangle XAB is isosceles. Prove that all the vertices of P lie on some circle center X.

Is it possible to place the digits 0, 1, 2 into unit squares of 100 x 100 cross- lined paper such that every 3 x 4 (and every 4 x 3) rectangle contains three 0s, four 1s and five 2s?

x₁, x₂, ..., xₙ are positive reals with sum 1. Let s be the largest of x₁/(1 + x₁), x₂/(1 + x₁ + x₂), ..., xₙ/(1 + x₁ + ... + xₙ). What is the smallest possible value of s? What are the corresponding xᵢ?

n teams compete in a tournament. Each team plays every other team once. In each game a team gets 2 points for a win, 1 for a draw and 0 for a loss. Given any subset S of teams, one can find a team (possibly in S) whose total score in the games with teams in S was odd. Prove that n is even.