8th Brasil 1986


1. 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.
2. Find the number of ways that a positive integer n can be represented as a sum of one or more consecutive positive integers.
3. 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, π).
4. Find all 10 digit numbers a₀a₁...a₉ such that for each k, a_k is the number of times that the digit k appears in the number.
5. 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.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.