International Mathematical Olympiad · 2019

Problems

Problem 1Let $\mathbb{Z}$ be the set of integers. Determine all functions $f\colon\mathbb{Z}\to\mathbb{Z}$ such that, for all integers $a$ and $b$ , $f(2a) + 2f(b) = f(f(a + b)).$ Solutions: 1
Problem 2In triangle $ABC$ , point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$ . Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$ , respectively, such that $PQ$ is parallel to $AB$ . Let $P_1$ be a point on line $PB_1$ , such that $B_1$ lies strictly between $P$ and $P_1$ , and $\angle PP_1C=\angle BAC$ . Similarly, let $Q_1$ be the point on line $QA_1$ , such that $A_1$ lies strictly between $Q$ and $Q_1$ , and $\angle CQ_1Q=\angle CBA$ . Prove that points $P,Q,P_1,Q_1$ are concyclic.Solutions: 1
Problem 3A social network has $2019$ users, some pairs of whom are friends (friendship is a symmetric relation). Events of the following kind may happen repeatedly, one at a time: three users $A$ , $B$ , and $C$ such that $A$ is friends with both $B$ and $C$ , but $B$ and $C$ are not friends, change their friendship statuses such that $B$ and $C$ are now friends, but $A$ is no longer friends with $B$ , and no longer friends with $C$ . All other friendship statuses remain unchanged. Initially, $1010$ users have $1009$ friends each, and $1009$ users have $1010$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.Solutions: 1
Problem 4Find all pairs $(k,n)$ of positive integers such that $k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}).$ Solutions: 1
Problem 5The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$ , then he turns over the $k$ th coin from the left; otherwise all coins show $T$ and he stops. For example, if $n=3$ , the process starting with $THT$ is $THT\to HHT\to HTT\to TTT$ , which stops after three operations. (a) Show that, for each initial configuration, Harry stops after a finite number of operations. (b) For each initial configuration $C$ , let $L(C)$ be the number of operations before Harry stops. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations.Solutions: 1
Problem 6Let $I$ be the incenter of acute triangle $ABC$ with $AB\ne AC$ . The incircle $\omega$ of $ABC$ is tangent to $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ again at $R$ (other than $D$ ). Line $AR$ meets $\omega$ again at $P$ (other than $R$ ). The circumcircles of triangles $PCE$ and $PBF$ meet again at $Q$ (other than $P$ ). Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$ .Solutions: 1