Let $ABC$ be an acute-angled triangle with $AC > AB$ and let $D$ be the foot of the $A$-angle bisector on $BC$. The reflections of lines $AB$ and $AC$ in line $BC$ meet $AC$ and $AB$ at points $E$ and $F$ respectively. A line through $D$ meets $AC$ and $AB$ 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 $\triangle EDG$ and $\triangle FDH$ are tangent to each other.

Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . < a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions: $0 \le b_i \le n$ for each $1 \le i \le k$, all the positive $b_i$ are distinct, the sums $a_i + b_i$, $1 \le i \le k$, 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 $a > b^2$. Proposed by Tynyshbek Anuarbekov, Kazakhstan

Let $\mathbb{R}^+ = (0, \infty)$ be the set of all positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0) = 0$ which satisfy the equality $f(f(x) + P(y)) = f(x - y) + 2y$ for all real numbers $x > y > 0$. Proposed by Sardor Gafforov, Uzbekistan