In a circle with a radius of $1$ is an inscribed hexagon (convex). Prove that if the multiple of all diagonals that connects vertices of neighboring sides is equal to $27$ then all angles of hexagon are equals. V. Petkov, I. Tonov

Find all polynomials $p(x)$ satisfying the condition: $$p(x^2-2x)=p(x-2)^2.$$

In the space is given a tetrahedron with length of the edge $2$. Prove that distances from some point $M$ to all of the vertices of the tetrahedron are integer numbers if and only if $M$ is a vertex of tetrahedron. J. Tabov

Let $0<x_1\le x_2\le\ldots\le x_n$. Prove that $$\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots+\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}\ge\frac{x_2}{x_1}+\frac{x_3}{x_2}+\ldots+\frac{x_n}{x_{n-1}}+\frac{x_1}{x_n}$$ I. Tonov

It is given a tetrahedron $ABCD$ and a plane $\alpha$ intersecting the three edges passing through $D$. Prove that $\alpha$ divides the surface of the tetrahedron into two parts proportional to the volumes of the bodies formed if and only if $\alpha$ is passing through the center of the inscribed tetrahedron sphere.

It is given a plane with a coordinate system with a beginning at the point $O$. $A(n)$, when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$. (a) Find $$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$(b) For which $\beta$ $(1<\beta<2)$ does the following limit exist? $$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$