BALKAN 2024

Let $A B C$ be an acute-angled triangle with $A C>A B$ and let $D$ be the foot of the $A$-angle bisector on $B C$. The reflections of lines $A B$ and $A C$ in line $B C$ meet $A C$ and $A B$ at points $E$ and $F$ respectively. A line through $D$ meets $A C$ and $A B$ at $G$ and $H$ respectively such that $G$ lies strictly between $A$ and $C$ while $H$ lies strictly between $B$ and $F$. Prove that the circumcircles of $\triangle E D G$ and $\triangle F D H$ are tangent to each other.

Let $n \geq k \geq 3$ be integers. Show that for every integer sequence $1 \leq a_{1}<a_{2}<\ldots<$ $a_{k} \leq n$ one can choose non-negative integers $b_{1}, b_{2}, \ldots, b_{k}$, satisfying the following conditions: (i) $0 \leq b_{i} \leq n$ for each $1 \leq i \leq k$, (ii) all the positive $b_{i}$ are distinct, (iii) the sums $a_{i}+b_{i}, 1 \leq i \leq k$, form a permutation of the first $k$ terms of a non-constant arithmetic progression.

Let $a$ and $b$ be distinct positive integers such that $3^{a}+2$ is divisible by $3^{b}+2$. Prove that $a>b^{2}$ 。

Let $\mathbb{R}^{+}=(0, \infty)$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0)=0$ which satisfy the equality

$$f(f(x)+P(y))=f(x-y)+2 y$$

for all real numbers $x>y>0$.