MK-MO 2017

Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each natural integer $n>1$ and for all $x,y \in \mathbb{N}$ the following holds: $$f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k$$

Find all natural integers $n$ such that $(n^3 + 39n - 2)n! + 17\cdot 21^n + 5$ is a square.

Let $x,y,z \in \mathbb{R}$ such that $xyz = 1$ . Prove that: $$\left(x^4 + \frac{z^2}{y^2}\right)\left(y^4 + \frac{x^2}{z^2}\right)\left(z^4 + \frac{y^2}{x^2}\right) \ge \left(\frac{x^2}{y} + 1 \right)\left(\frac{y^2}{z} + 1 \right)\left(\frac{z^2}{x} + 1 \right).$$

Let $O$ be the circumcenter of the acute triangle $ABC$ ( $AB < AC$ ). Let $A_1$ and $P$ be the feet of the perpendicular lines drawn from $A$ and $O$ to $BC$ , respectively. The lines $BO$ and $CO$ intersect $AA_1$ in $D$ and $E$ , respectively. Let $F$ be the second intersection point of $\odot ABD$ and $\odot ACE$ . Prove that the angle bisector od $\angle FAP$ passes through the incenter of $\triangle ABC$ .

Let $n>1 \in \mathbb{N}$ and $a_1, a_2, ..., a_n$ be a sequence of $n$ natural integers. Let: $$b_1 = \left[\frac{a_2 + \cdots + a_n}{n-1}\right], b_i = \left[\frac{a_1 + \cdots + a_{i-1} + a_{i+1} + \cdots + a_n}{n-1}\right], b_n = \left[\frac{a_1 + \cdots + a_{n-1}}{n-1}\right]$$ Define a mapping $f$ by $f(a_1,a_2, \cdots a_n) = (b_1,b_2,\cdots,b_n)$ .

a) Let $g: \mathbb{N} \to \mathbb{N}$ be a function such that $g(1)$ is the number of different elements in $f(a_1,a_2, \cdots a_n)$ and $g(m)$ is the number od different elements in $f^m(a_1,a_2, \cdots a_n) = f(f^{m-1}(a_1,a_2, \cdots a_n)); m>1$ . Prove that $\exists k_0 \in \mathbb{N}$ s.t. for $m \ge k_0$ the function $g(m)$ is periodic.

b) Prove that $\sum_{m=1}^k \frac{g(m)}{m(m+1)} < C$ for all $k \in \mathbb{N}$ , where $C$ is a function that doesn't depend on $k$ .