BALKAN 2017

Find all ordered pairs $(x, y)$ of positive integers such that:

$$x^{3}+y^{3}=x^{2}+42 x y+y^{2} \text {. }$$

Let $A B C$ be an acute triangle with with $A B<A C$ and let $\Gamma$ be its circumcircle. Let the tangents to $\Gamma$ at $B$ and $C$ be $t_{B}$ and $t_{C}$ respectively and let their point of intersection be $L$. The line through $B$ parallel to $A C$ intersects $t_{C}$ at $D$. The line through $C$ parallel to $A B$ intersects $t_{B}$ at $E$. The circumcircle of triangle $B D C$ meets the side $A C$ at $T$ where $T$ lies between $A$ and $C$. The circumcircle of triangle $B E C$ meets the line $A B$ at $S$ where $B$ lies between $A$ and $S$. Prove that the lines $S T, B C$ and $A L$ are concurrent.

Let $\mathbb{N}$ be the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that:

$$n+f(m) \text { divides } f(n)+n f(m)$$

There are $n>2$ students sitting at a round table. Initially each student has exactly one candy. At each step, each student chooses one of the following operations: (a) Pass one candy to the student on their left or the student on their right. (b) Divide all their candies into two, possibly empty, sets and pass one set to the student on their left and the other to the student on their right.

At each step the students perform their chosen operations simultaneously. An arrangement of candies is legal if it can be obtained in a finite number of steps. Find the number of legal arrangements. (Two arrangements are different if there is a student who has different numbers of candies in each one.)