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.