2012 IMO Problems

Problems of the 53st IMO 2012 in Mar del Plata, Argentina.

Day 1

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 lines $AB$ 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 $C$ .)

Author: Evangelos Psychas, Greece

Solution

Problem 2.

Let ${{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}$ be positive real numbers that satisfy ${{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1$ . Prove that $\left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n\gneq n^n$

Author: Angelo di Pasquale, Australia

Solution

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 $1\le x\le N$ . Player $A$ keeps $x$ secret, and truthfully tells $N$ to 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 $S$ of positive integers (possibly one specified in some previous question), and asking $A$ whether $x$ belongs to $S$ . Player $B$ may ask as many such 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 $x$ belongs to $X$ , then $B$ wins; otherwise, he loses. Prove that:

If $n\ge {{2}^{k}}$ , then $B$ can guarantee a win.
For all sufficiently large $k$ , there exists an integer $n\ge {1.99^k}$ such that $B$ cannot guarantee a win.

Author: David Arthur, Canada

Solution

Day 2

Problem 4.

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ such that, for all integers $a$ , $b$ , $c$ that satisfy $a+b+c = 0$ , the following equality holds: $f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).$ (Here $\mathbb{Z}$ denotes the set of integers.)

Author: Liam Baker, South Africa

Solution

Problem 5.

Let $ABC$ be a triangle with $\angle BCA=90{}^\circ$ , and 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 $BK = BC$ . 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 $MK = ML$ .

Author: Josef Tkadlec, Czech Republic

Solution

Problem 6.

Find all positive integers n for which there exist non-negative integers $a_1$ , $a_2$ , $\ldots$ , $a_n$ such that $\frac{1}{{{2}^{{{a}_{1}}}}}+\frac{1}{{{2}^{{{a}_{2}}}}}+\cdots +\frac{1}{{{2}^{{{a}_{n}}}}}=\frac{1}{{{3}^{{{a}_{1}}}}}+\frac{2}{{{3}^{{{a}_{2}}}}}+\cdots +\frac{n}{{{3}^{{{a}_{n}}}}}=1$

Author: Dušan Djukić, Serbia

Solution

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2012 IMO Problems

Contents

Day 1

Problem 1.

Problem 2.

Problem 3.

Day 2

Problem 4.

Problem 5.

Problem 6.

Resources