Find all $c\in \mathbb{R}$ such that there exists a function $f:\mathbb{R}\to \mathbb{R}$ satisfying $$(f(x)+1)(f(y)+1)=f(x+y)+f(xy+c)$$for all $x,y\in \mathbb{R}$. Proposed by Kaan Bilge
2023 Turkey Olympic Revenge
4 March 2023 - Day 1
Let $ABC$ be a triangle. A point $D$ lies on line $BC$ and points $E,F$ are taken on $AC,AB$ such that $DE \parallel AB$ and $DF\parallel AC$. Let $G = (AEF) \cap (ABC) \neq A$ and $I = (DEF) \cap BC\neq D$. Let $H$ and $O$ denote the orthocenter and the circumcenter of triangle $DEF$. Prove that $A,O,I$ are collinear if and only if $G,H,I$ are collinear. Proposed by Kaan Bilge
Find all polynomials $P$ with integer coefficients such that $$s(x)=s(y) \implies s(|P(x)|)=s(|P(y)|).$$for all $x,y\in \mathbb{N}$. Note: $s(x)$ denotes the sum of digits of $x$. Proposed by Şevket Onur YILMAZ
5 March 2023 - Day 2
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all integers $x$ and $y$, the number $$f(x)^2+2xf(y)+y^2$$is a perfect square. Proposed by Barış Koyuncu
There are $10$ cups, each having $10$ pebbles in them. Two players $A$ and $B$ play a game, repeating the following in order each move: $\bullet$ $B$ takes one pebble from each cup and redistributes them as $A$ wishes. $\bullet$ After $B$ distributes the pebbles, he tells how many pebbles are in each cup to $A$. Then $B$ destroys all the cups having no pebbles. $\bullet$ $B$ switches the places of two cups without telling $A$. After finitely many moves, $A$ can guarantee that $n$ cups are destroyed. Find the maximum possible value of $n$. (Note that $A$ doesn't see the cups while playing.) Proposed by Emre Osman
In triangle $ABC$, $D$ is a variable point on line $BC$. Points $E,F$ are on segments $AC, AB$ respectively such that $BF=BD$ and $CD=CE$. Circles $(AEF)$ and $(ABC)$ meet again at $S$. Lines $EF$ and $BC$ meet at $P$ and circles $(PDS)$ and $(AEF)$ meet again at $Q$. Prove that, as $D$ varies, isogonal conjugate of $Q$ with respect to triangle $ ABC$ lies on a fixed circle. Proposed by Serdar Bozdag