IMO 2012

Official IMO 2012 problems  ·  0/6 solved.

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

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

Author: Angelo di Pasquale, Australia

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:

1. If $n\ge {{2}^{k}}$, then $B$ can guarantee a win.
2. 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

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

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

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