IMO Problems 2009

IMO Problems 2009

Day 1

IMO 2009 Problem 1

Let n be a positive integer and let a₁, a₂, a₃, …, aₖ (k ≥ 2) be distinct integers in the set {1, 2, 3, …, n} such that n divides aᵢ(aᵢ₊₁ – 1) for i = 1, 2, …, k – 1.
Prove that n does not divide aₖ(a₁ – 1).

IMO 2009 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 a tangent to the circle Γ.
Prove that OP = OQ.

IMO 2009 Problem 3

Suppose that s₁, s₂, s₃, … is a strictly increasing sequence of positive integers such that the subsequences ss₁, ss₂, ss₃, … and ss₁+1, ss₂+1, ss₃+1, … are both arithmetic progressions.

Prove that the sequence s₁, s₂, s₃, … itself is an arithmetic progression.

Day 2

IMO 2009 Problem 4

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

IMO 2009 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 exist and non degenerate triangle with sides of lengths
a, f(b), f(b + f(a) – 1).
Note: A triangle is non degenerate if its vertices are not collinear.

IMO 2009 Problem 6

Let a₁, a₂, a₃, …, aₙ positive integers and let M be a set of n – 1 positive integers not containing s = a₁ + a₂ + a₃ + … + aₙ. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with length a₁, a₂, a₃, …, 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.