OBM 1986

A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times.

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

The Poincare plane is a half-plane bounded by a line $R$ . The lines are taken to be (1) the half-lines perpendicular to $R$ , and (2) the semicircles with center on $R$ . Show that given any line $L$ and any point $P$ not on $L$ , there are infinitely many lines through $P$ which do not intersect $L$ . Show that if $ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$ .

Find all $10$ digit numbers $a_0a_1...a_9$ such that for each $k, a_k$ is the number of times that the digit $k$ appears in the number.

A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$ , then there is a number equal to $N$ in the border squares.