IMO 2019

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$ , $$f(2a)+2f(b)=f(f(a+b)).$$ Proposed by Liam Baker, South Africa

In 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 $P,Q,P_1$ , and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine

A social network has $2019$ users, some pairs of whom are friends. Whenever user $A$ is friends with user $B$ , user $B$ is also friends with user $A$ . 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 are 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.

Proposed by Adrian Beker, Croatia

Find all pairs $(k,n)$ of positive integers such that $$k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}).$$ Proposed by Gabriel Chicas Reyes, El Salvador

The 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 the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$ , 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. For example, $L(THT) = 3$ and $L(TTT) = 0$ . Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$ .

Proposed by David Altizio, USA

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$ . The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$ , and $AB$ at $D, E,$ and $F$ , respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$ . Line $AR$ meets $\omega$ again at $P$ . The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$ .

Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$ .

Proposed by Anant Mudgal, India