Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex $v_0$ and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the $i$-th turn the animal whose turn it is picks a vertex $v_i$ adjacent to $v_{i-1}$ that hasn't been picked before and eats all its apples. The game ends when there is no such vertex $v_i$. Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?

Triangle $\Delta ABC$ is inscribed in circle $\Omega$. The tangency point of $\Omega$ and the $A$-mixtilinear circle of $\Delta ABC$ is $T$. Points $E$, $F$ were chosen on $AC$, $AB$ respectively so that $EF\parallel BC$ and $(TEF)$ is tangent to $\Omega$. Let $\omega$ denote the $A$-excircle of $\Delta AEF$, which is tangent to sides $EF$, $AE$, $AF$ at $K$, $Y$, $Z$ respectively. Line $AT$ intersects $\omega$ at two points $P$, $Q$ with $P$ between $A$ and $Q$. Let $QK$ and $YZ$ intersect at $V$, and let the tangent to $\omega$ at $P$ and the tangent to $\Omega$ at $T$ intersect at $U$. Prove that $UV\parallel BC$.

Find all (weakly) increasing $f\colon \mathbb{R}\to \mathbb{R}$ for which \[f(f(x)+y)=f(f(y)+x)\]holds for all $x, y\in \mathbb{R}$.

Let $c$ be a positive real and $a_1, a_2, \dots$ be a sequence of nonnegative integers satisfying the following conditions for every positive integer $n$: (i)$\frac{2^{a_1}+2^{a_2}+\cdots+2^{a_n}}{n}$ is an integer; (ii)$\textbullet 2^{a_n}\leq cn$. Prove that the sequence $a_1, a_2, \dots$ is eventually constant.