IMO 2009 Germany - Day 1 Problems

Problem 1. Let n be a positive integer and let a1,a2,a3,...,ak ( k≥ 2) be distinct integer in the set { 1,2,...,n} such that n divides ai(ai+1-1) for i = 1,2,...,k-1. Prove that n does not divide ak(a1-1).

Problem 2. Let ABC be a triangle with circumcenter O. The points P and Q are interior points of the sides CA and AB respectively. Let K,L and M be the midpoints of the segments BP,CQ and PQ. respectively, and let Γ be the circle passing through K,L and M. Suppose that the line PQ is tangent to the circle Γ. Prove that OP = OQ.

Problem 3. Suppose that s1,s2,s3, ... is a strictly increasing sequence of positive integers such that the sub-sequences ss_{1},ss_{2},ss_{3}, ... and ss_{1+1},ss_{2+1},ss_{3+1}, ... are both arithmetic progressions. Prove that the sequence s1,s2,s3, ... is itself an arithmetic progression.