Here are the IMO'96 problems as posted by Carl Miller:
First day (4 1/2 hours; 7 points per problem)
1. Let ABCD be a rectangular board with |AB| = 20, |BC| = 12. The board
is divided into 20 x 12 unit squares. Let r be a given positive integer.
A coin can be moved from one square to another if and only if the distance
between the centres of the two squares is [root] r. The task is to find a
sequence of moves taking the coin from the square which has A as a vertex
to the square which has B as a vertex.
(a) Show that the task cannot be done if r is divisible by 2 or 3.
(b) Prove that the task can be done if r = 73.
(c) Can the task be done when r = 97?
2. Let P be a point inside triangle ABC such that
= p/2.
6. Let n, p, q be positive integers with n > p+q. Let x_0, x_1, ..., x_n
be integers satisfying the following conditions:
(a) x_0 = x_n = 0;
(b) for each integer i with 1 <= i <= n,
either x_i - x_(i-1) = p or x_i - x_(i-1) = -q.
Show that there eixsts a pair (i,j) of indices with i < j and (i,j) [not
equal to] (0,n) such that x_i = x_j.
The exam was quite hard this year... only one person got a 42 (from
Romania). Romania won with 187, the US was 2nd with 185, and the next
highest score was 167.