| # 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. | |