# 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 $\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n> n^n.$

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 (Du¹an Djukiæ, 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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