IMO Problems 2012

IMO Problems 2012

Day 1

IMO 2012 Problem 1

Given triangle ABC, the point J is the centre of the excircle opposite the vertex A. This excircle is tangent to the side BC at M and to the line AB and Ac at K and L respectively. The lines LM and BJ meet at F and the lines KM and CJ meet at G. Let S be the point of intersection of the lines AF and BC and let T be the point of intersection of the lines AG and BC.

Prove that M is the midpoint of ST. (The excircle of ABC opposite the vertex A is the circle that is tangent to the line segment BC, to the ray AB beyond B and to the ray AC beyond the C).

IMO 2012 Problem 2

Let n ≥ 3 be an integer and let a₂, a₃…, aₙ be positive real numbers such that a₂a₃..aₙ = 1.

Prove that (1 + a₂)²(1 + a₃)³ … (1 + aₙ)ⁿ ≥ nⁿ

IMO 2012 Problem 3

The liar’s guessing game is a game played between two players A and B. The rules of the game depend on two positive integers k and n which are known to Both players. At the start of the game A chooses integers x and N with 1 ≤ x ≤ N. Player A keeps x secret and truthfully tells N to the player B. Player B now tries to obtain information about x by asking player A questions as follows:

Each question consists of B specifying an arbitrary set S a positive integers (possibly one specified in some previous question) and asking A whether x belongs to S. Player B may ask as many questions as he wishes. After each question player A must immediately answer it with yes or no, but is allowed to lie as many times as she wants. The only restriction is that among any k + 1 consecutive answers, at least one answer must be truthful.

After B has ask as many questions as he wants, he must specify a set X of at most n positive integers. If x belongs to X, then B wins, otherwise he loses. Prove that:

(i) If n ≥ 2ᵏ, then B can guarantee a win.
(ii) For all sufficiently large k, there exists an integer n ≥ (1.99)ᵏ such B we cannot guarantee a win.

Day 2

IMO 2012 Problem 4

Find all functions f: Z → Z {Here Z denotes the set of integers} such that, for all integers a, b, and c satisfying a + b + c = 0, the following equality holds:

f²(a) + f²(b) + f²(c) = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).

IMO 2012 Problem 5

Let ABC be a triangle with angle(BCA) = 90ᵒ, let D be the foot of the altitude from C. Let X be a point in the interior of the segment CD. Let K be the point on the segment AX such that PK = BC. Similarly, let L be the point on the segment BX such that AL = AC. Let M be the point of intersection of AL and BK.

Show that MK = ML.

IMO 2012 Problem 6

Find all positive integers n for which there exist non negative integers a₁, a₂, a₃…, aₙ such that:

1/2a₁ + 1/2a₂ + … + 1/2aₙ = 1/3a₁ + 2/3a₂ + … + n/3aₙ = 1.