KZ-MO 2016

Prove that one can arrange all positive divisors of any given positive integer around a circle so that for any two neighboring numbers one is divisible by another.

Find all rational numbers $a$ ,for which there exist infinitely many positive rational numbers $q$ such that the equation $[x^a].{x^a}=q$ has no solution in rational numbers.(A.Vasiliev)

Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$ ,respectively. $AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$ .Prove that $\angle AXZ =\angle BXZ$ .(C.Ilyasov)

Attachments:

In isosceles triangle $ABC$ ( $CA=CB$ ), $CH$ is altitude and $M$ is midpoint of $BH$ .Let $K$ be the foot of the perpendicular from $H$ to $AC$ and $L=BK \cap CM$ .Let the perpendicular drawn from $B$ to $BC$ intersects with $HL$ at $N$ .Prove that $\angle ACB=2 \angle BCN$ .(M. Kunhozhyn)

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$ , and the sum of the lengths of the segments with the ends of different colors is $400$ . Prove that you can draw a straight line separating everything red dots from all blue ones.

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$ : $$a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$ ) such that $a_{m(C)}>C$ .