International Mathematical Olympiad · 2012

Problems

Problem 1Let $ABC$ be a triangle and $J$ the center of the $A$ -excircle. 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 $ST$ .Solutions: 1
Problem 2Let $n\ge3$ be an integer, and let $a_2,a_3,\dots,a_n$ be positive real numbers such that $a_2a_3\cdots a_n=1$ . Prove that $(1+a_2)^2(1+a_3)^3\cdots(1+a_n)^n>n^n$ .Solutions: 1
Problem 3In the liar's guessing game, A chooses an integer $x$ with $1\le x\le N$ and tells B the integer $N$ . B asks questions of the form “does $x$ belong to the set $D$ ?”, where $D$ is any set of positive integers. A may answer each question truthfully or falsely, but among every $k+1$ consecutive answers at least one must be truthful. After finitely many questions B names a set $X$ of at most $n$ positive integers and wins if $x\in X$ . Prove (a) if $n\ge2^k$ then B can guarantee a win; (b) for all sufficiently large $k$ , some $n\ge1.99^k$ defeats every strategy.Solutions: 1
Problem 4Find all functions $f:\mathbb Z\to\mathbb Z$ such that for all integers $a,b,c$ with $a+b+c=0$ , $f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$ .Solutions: 1
Problem 5Let $ABC$ be a triangle with $\angle BCA=90^\circ$ , and let D be the foot of the altitude from C. Let X be interior to CD. Let K lie on AX with $BK=BC$ , and L lie on BX with $AL=AC$ . Let $M=AL\cap BK$ . Prove $MK=ML$ .Solutions: 1
Problem 6Find all positive integers $n$ for which there exist nonnegative integers $a_1,\dots,a_n$ such that $\sum_{i=1}^n2^{-a_i}=\sum_{i=1}^n i3^{-a_i}=1$ .Solutions: 1