RS-SMO 2020

Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$ .

We are given a polyhedron with at least $5$ vertices, such that exactly $3$ edges meet in each of the vertices. Prove that we can assign a rational number to every vertex of the given polyhedron such that the following conditions are met: $(i)$ At least one of the numbers assigned to the vertices is equal to $2020$ . $(ii)$ For every polygonal face, the product of the numbers assigned to the vertices of that face is equal to $1$ .

We are given a triangle $ABC$ . Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$ , with the arrangment of points $D - A - B - E$ . The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$ , and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$ . Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$ .

In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$ , the diagonals $AC$ and $BD$ intersect at the point $E$ . Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$ . Prove that the lines $AP$ , $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.

For a natural number $n$ , with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$ . Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$ . $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$ .

Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$ .

We are given a natural number $k$ . Let us consider the following game on an infinite onedimensional board. At the start of the game, we distrubute $n$ coins on the fields of the given board (one field can have multiple coins on itself). After that, we have two choices for the following moves: $(i)$ We choose two nonempty fields next to each other, and we transfer all the coins from one of the fields to the other. $(ii)$ We choose a field with at least $2$ coins on it, and we transfer one coin from the chosen field to the $k-\mathrm{th}$ field on the left , and one coin from the chosen field to the $k-\mathrm{th}$ field on the right. $\mathbf{(a)}$ If $n\leq k+1$ , prove that we can play only finitely many moves. $\mathbf{(b)}$ For which values of $k$ we can choose a natural number $n$ and distribute $n$ coins on the given board such that we can play infinitely many moves.