BALKAN 2020

Let $A B C$ be an acute triangle with $A B=A C$, let $D$ be the midpoint of the side $A C$, and let $\gamma$ be the circumcircle of the triangle $A B D$. The tangent of $\gamma$ at $A$ crosses the line $B C$ at $E$. Let $O$ be the circumcentre of the triangle $A B E$. Prove that the midpoint of the segment $A O$ lies on $\gamma$.

Denote $\mathbb{Z}_{>0}=\{1,2,3, \ldots\}$ the set of all positive integers. Determine all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$, i) $\sum_{k=1}^{n} f(k)$ is a perfect square, and ii) $f(n)$ divides $n^{3}$.

Let $k$ be a positive integer. Determine the least integer $n \geqslant k+1$ for which the game below can be played indefinitely:

Consider $n$ boxes, labelled $b_{1}, b_{2}, \ldots, b_{n}$. For each index $i$, box $b_{i}$ contains initially exactly $i$ coins. At each step, the following three substeps are performed in order: (1) Choose $k+1$ boxes; (2) Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_{i}$, a number of $i$ coins. (3) If one of the boxes is left empty, the game ends; otherwise, go to the next step.

Let $a_{1}=2$ and, for every positive integer $n$, let $a_{n+1}$ be the smallest integer strictly greater than $a_{n}$ that has more positive divisors than $a_{n}$. Prove that $2 a_{n+1}=3 a_{n}$ only for finitely many indices $n$.