Point $P$ lies inside the triangle $ABC$. Points $K, L, M$ are symmetrics of point $P$ wrt the midpoints of the sides $BC, CA, AB$. Prove that the straight $AK, BL, CM$ intersect at one point.

Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.

Different points $A, B, C, D$ lie on a circle with a center at the point $O$ at such way that $\angle AOB$ $= \angle BOC =$ $\angle COD =$ $60^o$. Point $P$ lies on the shorter arc $BC$ of this circle. Points $K, L, M$ are projections of $P$ on lines $AO, BO, CO$ respectively . Show that (a) the triangle $KLM$ is equilateral, (b) the area of triangle $KLM$ does not depend on the choice of the position of point $P$ on the shorter arc $BC$

Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.

Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$. Show that the number $\frac12(c - a) (c - b)$ is square of an integer.

There are a lot of different real numbers written on the board. It turned out that for each two numbers written, their product was also written. What is the largest possible number of numbers written on the board?

On the circle $k$, the points $A,B$ are given, while $AB$ is not the diameter of the circle $k$. Point $C$ moves along the long arc $AB$ of circle $k$ so that the triangle $ABC$ is acute. Let $D,E$ be the feet of the altitudes from $A, B$ respectively. Let $F$ be the projection of point $D$ on line $AC$ and $G$ be the projection of point $E$ on line $BC$. (a) Prove that the lines $AB$ and $FG$ are parallel. (b) Determine the set of midpoints $S$ of segment $FG$ while along all allowable positions of point $C$.

Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.

A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, $CP = 3$. Determine the length of the segment $AP$.

Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.

The $8 \times 8$ board is covered with the same shape as in the picture to the right (each of the shapes can be rotated $90^o$) so that any two do not overlap or extend beyond the edge of the chessboard. Determine the largest possible number of fields of this chessboard can be covered as described above.