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

$$f(x+yf(x))+f(xy)=f(x)+f(2019y)f(x+y\; f(x))+f(x\; y)=f(x)+f(2019\; y)$$

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

$$\bigcup _{s\in S}\{s,Q(s),Q(Q(s)),Q(Q(Q(s))),\dots \}\backslash bigcup\_\{s\; \backslash in\; S\}\backslash \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \backslash ldots\backslash \}$$

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.