International Mathematical Olympiad · 2010

Problems

Problem 1Find all functions $f:\mathbb R\to\mathbb R$ such that, for all real numbers $x,y$ , $f(\lfloor x\rfloor y)=f(x)\lfloor f(y)\rfloor$ .Solutions: 1
Problem 2Let $I$ be the incenter of a triangle $ABC$ and let $\Gamma$ be its circumcircle. Let the line $AI$ intersect $\Gamma$ again at $D$ . Let $E$ be a point on the arc $BDC$ (the arc not containing $A$ ) and $F$ a point on the side $BC$ such that $\angle BAF=\angle CAE<\dfrac12\angle BAC$ . Let $G$ be the midpoint of the segment $IF$ . Prove that the lines $DG$ and $EI$ intersect on $\Gamma$ .Solutions: 1
Problem 3Find all functions $g:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that $\left(g(m)+n\right)\left(g(n)+m\right)$ is a perfect square for all $m,n\in\mathbb{Z}_{>0}$ .Solutions: 1
Problem 4Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$ ). The lines $AP$ , $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$ , $L$ , respectively $M$ . The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$ . Show that from $SC = SP$ follows $MK = ML$ .Solutions: 1
Problem 5Each of the six boxes $B_1,B_2,B_3,B_4,B_5,B_6$ initially contains one coin. A type 1 operation chooses a nonempty box $B_j$ with $1\le j\le5$ , removes one coin from it, and adds two coins to $B_{j+1}$ . A type 2 operation chooses a nonempty box $B_k$ with $1\le k\le4$ , removes one coin from it, and exchanges the contents of (possibly empty) boxes $B_{k+1}$ and $B_{k+2}$ . Determine whether a finite sequence of operations can leave $B_1,B_2,B_3,B_4,B_5$ empty and $B_6$ containing exactly $2010^{2010^{2010}}$ coins. Here $a^{b^c}$ means $a^{(b^c)}$ .Solutions: 1
Problem 6Let $a_1,a_2,a_3,\ldots$ be a sequence of positive real numbers. Suppose that for some positive integer $s$ , we have $a_n=\max\{a_k+a_{n-k}\mid 1\le k\le n-1\}$ for all $n>s$ . Prove that there exist positive integers $\ell$ and $N$ , with $\ell\le s$ , such that $a_n=a_\ell+a_{n-\ell}$ for all $n\ge N$ .Solutions: 1