International Mathematical Olympiad · 2015

Problems

Problem 1We say that a finite set $S$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $S$ , there is a point $C$ in $S$ such that $AC=BC$ . We say that $S$ is center-free if for any three different points $A$ , $B$ , $C$ in $S$ , there is no point $P$ in $S$ such that $PA=PB=PC$ . (a) Show that for all integers $n\ge3$ , there exists a balanced set consisting of $n$ points. (b) Determine all integers $n\ge3$ for which there exists a balanced center-free set consisting of $n$ points.Solutions: 1
Problem 2Determine all triples $(a,b,c)$ of positive integers such that each of the numbers $ab-c$ , $bc-a$ , and $ca-b$ is a power of $2$ (a power of $2$ is an integer of the form $2^n$ , where $n$ is a nonnegative integer).Solutions: 1
Problem 3Let $ABC$ be an acute triangle with $AB>AC$ . Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$ . Let $M$ be the midpoint of $BC$ . Let $Q$ be the point on $\Gamma$ such that $\angle HQA=90^\circ$ , and let $K$ be the point on $\Gamma$ such that $\angle HKQ=90^\circ$ . Assume that the points $A$ , $B$ , $C$ , $K$ , $Q$ are all different and lie on $\Gamma$ in this order. Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.Solutions: 1
Problem 4Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$ . A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$ , such that $B$ , $D$ , $E$ , $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$ , such that $A$ , $F$ , $B$ , $C$ , $G$ lie on $\Omega$ in this order. Let $K$ be the second intersection point of the circumcircle of triangle $BDF$ with segment $AB$ , and let $L$ be the second intersection point of the circumcircle of triangle $CGE$ with segment $AC$ . Suppose that the lines $FK$ and $GL$ are distinct and intersect at the point $X$ . Prove that $X$ lies on the line $AO$ .Solutions: 1
Problem 5Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the equation $f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$ for all real numbers $x$ and $y$ .Solutions: 1
Problem 6The sequence $a_1,a_2,\dots$ of integers satisfies the conditions: (i) $1\le a_j\le 2015$ for all $j\ge1$ ; (ii) $k+a_k\ne \ell+a_\ell$ for all $1\le k<\ell$ . Prove that there exist two positive integers $b$ and $N$ for which $\left|\sum_{j=m+1}^{n}(a_j-b)\right|\le 1007^2$ for all integers $m$ and $n$ such that $n>m\ge N$ .Solutions: 1