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News: IMO Day 1 (25 July 2007)

The first IMO 2007 paper was sat in Hanoi on 25 July 2007; the problems and the countries submitting them are shown.

Real numbers a₁, a₂, …, a_n are given. For each i (1 ≤ i ≤ n) define

d_i = max { a_j : 1 ≤ j ≤ i } − min { a_j : i ≤ j ≤ n }

and let

d = max { d_i : 1 ≤ i ≤ n }.

(a) Prove that, for any real numbers x₁ ≤ x₂ ≤ … ≤ x_n,

max { |x_i − a_i| : 1 ≤ i ≤ n } ≥ d / 2. (*)

(b) Show that there are real numbers x₁ ≤ x₂ ≤ … ≤ x_n such that equality holds in (*).

(New Zealand)
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 intersects line BC at G. Suppose also that EF = EG = EC. Prove that ℓ is the bisector of angle DAB.

(Luxembourg)
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 this competition, the largest size of a clique is even, prove that the competitors can be arranged in 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.

(Russia)