IMO Problems 2007

IMO Problems 2007

Day 1

IMO 2007 Problem 1

Real numbers a₁, a₂, …, aₙ are given, for each i (1 ≤ i ≤ n), define

dᵢ = Max{aⱼ : 1 ≤ j ≤ i} – Min{aⱼ : i ≤ j ≤ n} and let

d = Max{dᵢ : 1 ≤ i ≤ n}

(A) Prove that for any real numbers x₁ ≤ x₂ ≤ … ≤ xₙ

Max{|xᵢ – aᵢ| : 1 ≤ i ≤ n} ≥ d/2 (*)

(B) Show that there are real numbers x₁ ≤ x₂ ≤ … ≤ xₙ, such that the equality holds in (*).

IMO 2007 Problem 2

Consider five points A, B, C, D, and E such that ABCD is a parallelogram and BCED is a cyclic quadrilateral. Let L be a line passing through A. Suppose that L intersects the interior of the segment DC at F and intersect the line BC at G. Suppose also that EF = CG = EC. Prove that L is the bisector of angle DAB.

IMO 2007 Problem 3

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. In particular any group of fewer than two competitors is a clique. The number of members of a clique is called its size. Given that, in the competition, the largest size of a clique is even. Prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.

Day 2

IMO 2007 Problem 4

In a triangle ABC the bisector of angle BCA intersects the circumcircle again at R. The perpendicular bisector of BC at P and the perpendicular bisector of AC at Q. The midpoint of BC is K and the midpoint of AC is L. Prove that the Triangles RPK and RQL have the same area.

IMO 2007 Problem 5

Let a and b are positive integers. Show that if (4ab – 1) divides (4a² – 1)², then a = b.

IMO 2007 Problem 6

Let n be a positive integer, Consider

S = {(x, y, z) | x, y, z ∈ {1, 1, 2, …, n}, x + y + z > 0}

as a set of (n + 1)³ – 1 points in the three-dimensional space. Determine all smallest possible numbers of planes, the union of which contains S but does not include (0, 0, 0).