SILK-ROAD 2019

The altitudes of the acute-angled non-isosceles triangle $ABC$ intersect at the point $H$ . On the segment $C_1H$ , where $CC_1$ is the altitude of the triangle, the point $K$ is marked. Points $L$ and $M$ are the feet of perpendiculars from point $K$ on straight lines $AC$ and $BC$ , respectively. The lines $AM$ and $BL$ intersect at $N$ . Prove that $\angle ANK = \angle HNL$ .

Let $a_1,$ $a_2,$ $\ldots,$ $a_ {99}$ be positive real numbers such that $ia_j + ja_i \ge i + j$ for all $1 \le i <j \le 99.$ Prove , that $(a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .

Find all pairs of $(a, n)$ natural numbers such that $\varphi (a ^ n + n) = 2 ^ n.$ ( $\varphi (n)$ is the Euler function, that is, the number of integers from $1$ up to $n$ , relative prime to $n$ )

The sequence $\{a_n \}$ is defined as follows: $a_0 = 1$ and ${a_n} = \sum \limits_ {k = 1} ^ {[\sqrt n]} {{a_ {n - {k ^ 2 }}}}$ for $n \ge 1.$ Prove that among $a_1, a_2, \ldots, a_ {10 ^ 6}$ there are at least $500$ even numbers. (Here, $[x]$ is the largest integer not exceeding $x$ .)