MK-MO 2018

Determine all natural numbers $n$ such that $9^n - 7$ can be represented as a product of at least two consecutive natural numbers.

Let $n$ be a natural number and $C$ a non-negative real number. Determine the number of sequences of real numbers $1, x_{2}, ..., x_{n}, 1$ such that the absolute value of the difference between any two adjacent terms is equal to $C$ .

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that: $$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y$$ for all real $x,y \in \mathbb{R}$ Proposed by Nikola Velov

Let $t_{k} = a_{1}^k + a_{2}^k +...+a_{n}^k$ , where $a_{1}$ , $a_{2}$ , ... $a_{n}$ are positive real numbers and $k \in \mathbb{N}$ . Prove that $$\frac{t_{5}^2 t_1^{6}}{15} - \frac{t_{4}^4 t_{2}^2 t_{1}^2}{6} + \frac{t_{2}^3 t_{4}^5}{10} \geq 0$$ Proposed by Daniel Velinov

Given is an acute $\triangle ABC$ with orthocenter $H$ . The point $H'$ is symmetric to $H$ over the side $AB$ . Let $N$ be the intersection point of $HH'$ and $AB$ . The circle passing through $A$ , $N$ and $H'$ intersects $AC$ for the second time in $M$ , and the circle passing through $B$ , $N$ and $H'$ intersects $BC$ for the second time in $P$ . Prove that $M$ , $N$ and $P$ are collinear.

Proposed by Petar Filipovski