In triangle $ABC$, the incircle touches sides $AB$ and $BC$ at points $P$ and $Q$, respectively. Median of triangle $ABC$ from vertex $B$ meets segment $P Q$ at point $R$. Prove that angle $ARC$ is obtuse.

Positive real numbers $a$,$b$,$c$ are given such that $abc=1$.Prove that $$2(a+b+c)+\frac{9}{(ab+bc+ca)^2}\geq7.$$

The sequences of natural numbers $p_n$ and $q_n$ are given such that $$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$Prove that $p_n$ and $q_m$ are coprime for any m and n.

Through point $ P $ inside triangle $ ABC $, straight lines were drawn, parallel to the sides, until they intersect with the sides. In the three resulting parallelograms, diagonals that do not contain point $ P $, are drawn. Points $ A_1 $, $ B_1 $ and $ C_1 $ are the intersection points of the lines containing these diagonals such that $A_1$ and $A$ are in different sides of line $BC$ and $B_1$ and $C_1$ are similar. Prove that if hexagon $ AC_1BA_1CB_1 $ is inscribed and convex, then point $ P $ is the orthocenter of triangle $ A_1B_1C_1 $.

A finite family of finite sets $F$ is given, satisfying two conditions: (i) if $A, B \in F$, then $A \cup B \in F$; (ii) if $A \in F$, then the number of elements $| A |$ is not a multiple of $3$. Prove that you can specify at most two elements so that every set of the family $F$ contains at least one of them.

Find all functions $f:\mathbb{R^+}\to\mathbb{R^+}$ satisfying$$f(xy+x+y)=(f(x)-f(y))f(y-x-1)$$for all $x>0, y>x+1$.

Given a natural number $k>1$. Find the smallest number $\alpha$ satisfying the following condition. Suppose that the table $(2k + 1) \times (2k + 1)$ is filled with real numbers not exceeding $1$ in absolute value, and the sums of the numbers in all lines are equal to zero. Then you can rearrange the numbers so that each number remains in its row and all the sums over the columns will be at most $\alpha$.

A natural number $k>1$ is given. The sum of some divisor of $k$ and some divisor of $k - 1$ is equal to $a$,where $a>k + 1$. Prove that at least one of the numbers $a - 1$ or $a + 1$ composite.

In a convex quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $P$, and rays $BC$ and $AD$ at point $Q$. There is a point $T$ on the diagonal $AC$ such that the triangles $BTP$ and $DTQ$ are similar, in that order. Prove that $BD \Vert PQ$.

A regular hexagon with a side of $50$ was divided to equilateral triangles with unit side, parallel to the sides of the hexagon. It is allowed to delete any three nodes of the resulting lattice defining a segment of length $2$. As a result of several such operations, exactly one node remains. How many ways is this possible?

Positive real numbers $a,b,c$ are given such that $abc=1$. Prove that$$a+b+c+\frac{3}{ab+bc+ca}\geq4.$$

Triangle $ABC$ in which $AB <BC$, is inscribed in a circle $\omega$ and circumscribed about a circle $\gamma$ with center $I$. The line $\ell$ parallel to $AC$, touches the circle $\gamma$ and intersects the arcs $BAC$ and $BCA$ at points $P$ and $Q$, respectively. It is known that $PQ = 2BI$. Prove that $AP + 2PB = CP$.

A graph is called $7-chip$ if it obtained by removing at most three edges that have no vertex in common from a complete graph with seven vertices. Consider a complete graph $G$ with $v$ vertices which each edge of its is colored blue or red. Prove that there is either a blue path with $100$ edges or a red $7-chip$.

A set is called $six\ square$ if it has six pair-wise coprime numbers and for any partition of it into two set with three elements, the sum of the numbers in one of them is perfect square. Prove that there exist infinitely many $six\ square$.

There are $n$ triangles inscribed in a circle and all $3n$ of their vertices are different. Prove that it is possible to put a boy in one of the vertices in each triangle, and a girl in the other, so that boys and girls alternate on a circle.