IMO 2009

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.

Problem 4. Let ABC be a triangle with AB = AC. The angle bisectors of CAB and ABC meet the sides BC and CA at D and E, respectively. Let K be the incentre of triangle ADC. Suppose that angle BEK = 45. Find all possible values of angle CAB.

Problem 5. Determine all functions f from the set of positive integers to the set of positive integers such that, for all positive integers a and b, there exists a non-degenerate triangle with sides of lengths a, f(b), abd f(b+f(a)-1).

Problem 6. Let a1, a2, ..., an be distinct positive integers and let M be a set of positive integers not containing s = a1 + a2 + ... + an. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1, a2, ..., an in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M.