KZ-MO 2019

Prove for any positives $a,b,c$ the inequality $$\sqrt[3]{\dfrac{a}{b}}+\sqrt[5]{\dfrac{b}{c}}+\sqrt[7]{\dfrac{c}{a}}>\dfrac{5}{2}$$

The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least $\sqrt{2}$ . It is known that a regular triangle with side lenght $3$ cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?

Find all positive integers $n,k,a_1,a_2,...,a_k$ so that $n^{k+1}+1$ is divisible by $(na_1+1)(na_2+1)...(na_k+1)$

Given a checkered rectangle of size n × m. Is it always possible to mark $3$ or $4$ nodes of a rectangle so that at least one of the marked nodes lay on each straight line containing the side of the rectangle, and the non-self-intersecting polygon with vertices at these nodes has an area equal to $$\dfrac{1}{2}\min \left ( \text{gcd}(n, m), \dfrac{n+m}{\text{gcd}(n, m)} \right)$$ ?

The tangent line $l$ to the circumcircle of an acute triangle $ABC$ intersects the lines $AB, BC$ , and $CA$ at points $C', A'$ and $B'$ , respectively. Let $H$ be the orthocenter of a triangle $ABC$ . On the straight lines A'H, B′H and C'H, respectively, points $A_1, B_1$ and $C_1$ (other than $H$ ) are marked such that $AH = AA_1, BH = BB_1$ and $CH = CC_1$ . Prove that the circumcircles of triangles $ABC$ and $A_1B_1C_1$ are tangent.