We denote by $\mathbb{R}^+$ the set of all positive real numbers. Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property: \[f(x)f(y)=2f(x+yf(x))\] for all positive real numbers $x$ and $y$. Proposed by Nikolai Nikolov, Bulgaria

In a park there are 23 trees $t_0,t_1,\dots,t_{22}$ in a circle and 22 birds $b_1,n_2,\dots,b_{22}.$ Initially, each bird is in a tree. Every minute, the bird $b_i, 1\leqslant i\leqslant 22$ flies from the tree $t_j{}$ to the tree $t_{i+j}$ in clockwise order, indices taken modulo 23. Prove that there exists a moment when at least 6 trees are empty.

Given acute angle triangle $ ABC$. Let $ CD$be the altitude , $ H$ be the orthocenter and $ O$ be the circumcenter of $ \triangle ABC$ The line through point $ D$ and perpendicular with $ OD$ , is intersect $ BC$ at $ E$. Prove that $ \angle DHE = \angle ABC$.

Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$

Define sequence $a_{0}, a_{1}, a_{2}, \ldots, a_{2018}, a_{2019}$ as below: $ a_{0}=1 $ $a_{n+1}=a_{n}-\frac{a_{n}^{2}}{2019}$, $n=0,1,2, \ldots, 2018$ Prove $a_{2019} < \frac{1}{2} < a_{2018}$

Given a graph with $99$ vertices and degrees in $\{81,82,\dots,90\}$, prove that there exist $10$ vertices of this graph with equal degrees and a common neighbour. Proposed by Alireza Alipour

$AL$ is internal bisector of scalene $\triangle ABC$ ($L \in BC$). $K$ is chosen on segment $AL$. Point $P$ lies on the same side with respect to line $BC$ as point $A$ such that $\angle BPL = \angle CKL$ and $\angle CPL = \angle BKL$. $M$ is midpoint of segment $KP$, and $D$ is foot of perpendicular from $K$ on $BC$. Prove that $\angle AMD = 180^\circ - |\angle ABC - \angle ACB|$. Proposed by Mykhailo Shtandenko and Fedir Yudin

Find the greatest positive integer $n$ such that there exist positive integers $a_1, a_2, ..., a_n$ for which the following holds $a_{k+2} = \dfrac{(a_{k+1}+a_k)(a_{k+1}+1)}{a_k}$ for all $1 \le k \le n-2$. Proposed by Mykhailo Shtandenko and Oleksii Masalitin