Circle touches parallelogram‘s $ABCD$ borders $AB, BC$ and $CD$ respectively at points $K, L$ and $M$. Perpendicular is drawn from vertex $C$ to $AB$ . Prove, that the line $KL$ divides this perpendicular into two equal parts (with the same length).

Finite set $A$ has such property: every six its distinct elements’ sum isn’t divisible by $6$. Does there exist such set $A$ consisting of $11$ distinct natural numbers?

Given such positive real numbers $a, b$ and $c$, that the system of equations: $ \{\begin{matrix}a^2x+b^2y+c^2z=1&&\\xy+yz+zx=1&&\end{matrix} $ has exactly one solution of real numbers $(x, y, z)$. Prove, that there is a triangle, which borders lengths are equal to $a, b$ and $c$.

(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$? b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?

Given real numbers $x$ and $y$. Let $s_{1}=x+y, s_{2}=x^2+y^2, s_{3}=x^3+y^3, s_{4}=x^4+y^4$ and $t=xy$. a) Prove, that number $t$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers. b) Prove, that number $s_{1}$ is rational, if $s_{2}, s_{3}$ and $s_{4}$ are rational numbers. c) Can number $s_{1}$ be irrational, if $s_{2}$ and $s_{3}$ are rational numbers?

Circles ω1 and ω2 have no common point. Where is outerior tangents a and b, interior tangent c. Lines a, b and c touches circle ω1 respectively on points A1, B1 and C1, and circle ω2 – respectively on points A2, B2 and C2. Prove that triangles A1B1C1 and A2B2C2 area ratio is the same as ratio of ω1 and ω2 radii.