KZ-MO 2017

The non-isosceles triangle $ABC$ is inscribed in the circle $\omega$ . The tangent line to this circle at the point $C$ intersects the line $AB$ at the point $D$ . Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$ , respectively. The point $M$ is on the side $AB$ such that $\frac{AK}{BL} = \frac{AM}{BM}$ . Let the perpendiculars from the point $M$ to the straight lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$ respectively. Prove that $2\angle CQP=\angle ACB$

For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$ , prove this inequality holds $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$

$\{a_n\}$ is an infinite, strictly increasing sequence of positive integers and $a_{a_n}\leq a_n+a_{n+3}$ for all $n\geq 1$ . Prove that, there are infinitely many triples $(k,l,m)$ of positive integers such that $k<l<m$ and $a_k+a_m=2a_l$

The acute triangle $ABC$ $(AC> BC)$ is inscribed in a circle with the center at the point $O$ , and $CD$ is the diameter of this circle. The point $K$ is on the continuation of the ray $DA$ beyond the point $A$ . And the point $L$ is on the segment $BD$ $(DL> LB)$ so that $\angle OKD = \angle BAC$ , $\angle OLD = \angle ABC$ . Prove that the line $KL$ passes through the midpoint of the segment $AB$ .

Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$ . We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$ . What is the largest number of sets that can be written on the board, so that any set is not more than the other set?

Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$