RMM 2019

Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^{2}$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^{k}$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob's win?

Russia, Maxim Didin

Let $A B C D$ be an isosceles trapezoid with $A B \| C D$. Let $E$ be the midpoint of $A C$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $A B E$ and $C D E$, respectively. Let $P$ be the crossing poitn of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $P E$ is tangent to $\Omega$.

Slovenia, Jakob Jurij Snou

Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least $(1+\varepsilon) v$ edges has two distinct simple cycles of equal lengths. (Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.)

Russia, Fedor Petrov

Prove that for every positive integer $n$ there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly $n$ different triangulations. (A triangulation is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon.)

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying

$$f(x+y f(x))+f(x y)=f(x)+f(2019 y)$$

for all real numbers $x$ and $y$.

Find all pairs of integers $(c, d)$, both greater than 1 , such that the following holds:

For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p>c(2 c+1)$, there exists a set $S$ of at most $\left(\frac{2 c-1}{2 c+1}\right) p$ integers, such that

$$\bigcup_{s \in S}\{s, Q(s), Q(Q(s)), Q(Q(Q(s))), \ldots\}$$

is a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$ ).

Each of the three problems is worth 7 points. Time allowed $4 \frac{1}{2}$ hours.