International Mathematical Olympiad · 2018

Problems

Problem 1Let $\Gamma$ be the circumcircle of acute triangle $ABC$ . Points $D$ and $E$ are on segments $AB$ and $AC$ , respectively, such that $AD=AE$ . The perpendicular bisectors of $BD$ and $CE$ intersect the minor arcs $AB$ and $AC$ of $\Gamma$ at points $F$ and $G$ , respectively. Prove that lines $DE$ and $FG$ are either parallel or they are the same line.Solutions: 1
Problem 2Find all integers $n \ge 3$ for which there exist real numbers $a_1, a_2, \ldots, a_n$ satisfying $a_{n+1}=a_1$ , $a_{n+2}=a_2$ , and $a_ia_{i+1}+1=a_{i+2}$ for $i=1,2,\ldots,n$ .Solutions: 1
Problem 3An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. The following is a four-row anti-Pascal triangle containing every integer from 1 through 10: $\begin{array}{ccccccccccc} &&&&4&&&&\\ &&&2&&6&&&\\ &5&&7&&1&&&\\ 8&&3&&10&&9 \end{array}$ Does there exist an anti-Pascal triangle with 2018 rows which contains every integer from 1 to $1+2+\cdots+2018$ ?Solutions: 1
Problem 4A site is any point $(x,y)$ in the plane for which $x,y\in\{1,2,\ldots,20\}$ . Initially all 400 sites are unoccupied. Amy and Ben take turns placing stones on unoccupied sites, with Amy going first. Amy places a red stone only if the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$ . Ben places a blue stone on any unoccupied site, without any distance restriction. They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, regardless of how Ben plays.Solutions: 1
Problem 5Let $a_1,a_2,\ldots$ be an infinite sequence of positive integers. Suppose there is an integer $N>1$ such that, for every $n\ge N$ , the number $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\cdots+\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}$ is an integer. Prove that there is a positive integer $M$ such that $a_m=a_{m+1}$ for all $m\ge M$ .Solutions: 1
Problem 6A convex quadrilateral $ABCD$ satisfies $AB\cdot CD=BC\cdot DA$ . Point $X$ lies inside $ABCD$ so that $\angle XAB=\angle XCD$ and $\angle XBC=\angle XDA$ . Prove that $\angle BXA+\angle DXC=180^\circ$ .Solutions: 1