The 53rd International Mathematical Olympiad: Problems and Solutions

Day 1 (July 10th, 2012)

Problem 1 (Evangelos Psychas, Greece)
 

Given a triangle \( ABC \), let \( J \) be the center of the excircle opposite to the vertex \( A \). This circle is tangent to lines \( AB \), \( AC \), and \( BC \) at \( K \), \( L \), and \( M \), respectively. The lines \( BM \) and \( JF \) meet at \( F \), and the lines \( KM \) and \( CJ \) meet at \( G \). Let \( S \) be the intersection of \( AF \) and \( BC \), and let \( T \) be the intersection of \( AG \) and \( BC \). Prove that \( M \) is the midpoint of \( BC \).

Problem 2 (Angelo di Pasquale, Australia)
 

Let \( a_2 \), \( a_3 \), \( \dots \), \( a_n \) be positive real numbers that satisfy \( a_2\cdot a_3\cdots a_n=1 \). Prove that \[ \left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n> n^n.\]

Problem 3 (David Arthur, Canada)
 

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 the player \( A \) chooses integers \( x \) and \( N \) with \( 1\leq x\leq N \). Player \( A \) keeps \( x \) secret, and truthfully tells \( N \) to the player \( B \). The 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 \( S \) of 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 asked as many questions as he wants, he must specify a set \( X \) of at most \( n \) positive integers. If \( x\in X \), then \( B \) wins; otherwise, he loses. Prove that:

(a) If \( n\geq 2^k \) then \( B \) has a winning strategy.

(b) There exists a positive integer \( k_0 \) such that for every \( k\geq k_0 \) there exists an integer \( n\geq 1.99^k \) for which \( B \) cannot guarantee a victory.

Day 2 (July 11th, 2012)

Problem 4 (Liam Baker, South Africa)
 

Find all functions \( f:\mathbb Z\to\mathbb Z \) such that, for all integers \( a \), \( b \), \( c \) with \( a+b+c=0 \) the following equality holds: \[ f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a).\]

Problem 5 (Josef Tkadlec, Czech Republic)
 

Given a triangle \( ABC \), assume that \( \angle C=90^{\circ} \). Let \( D \) be the foot of the perpendicular from \( C \) to \( AB \), and let \( X \) be any point of the segment \( CD \). Let \( K \) and \( L \) be the points on the segments \( AX \) and \( BX \) such that \( BK=BC \) and \( AL=AC \), respectively. Let \( M \) be the intersection of \( AL \) and \( BK \).

Prove that \( MK=ML \).

Problem 6 (Dušan Djukić, Serbia)
 

Find all positive integers \( n \) for which there exist non-negative integers \( a_1 \), \( a_2 \), \( \dots \), \( a_n \) such that \[ \frac1{2^{a_1}}+\frac1{2^{a_2}}+\cdots+\frac1{2^{a_n}}=\frac1{3^{a_1}}+\frac2{3^{a_2}}+\cdots+\frac{n}{3^{a_n}}=1.\]




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