RMM/md/en-2019-RMM2019-Day2-English.md

# The $11^{\text {th }}$ Romanian Master of Mathematics Competition 

Day 2: Saturday, February 23, 2019, Bucharest

Language: English

Problem 4. 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.)

Problem 5. 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 $y$.
Problem 6. 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.