BG-MO 1974

Find all natural numbers n with the following property: there exists a permutation $(i_1,i_2,\ldots,i_n)$ of the numbers $1,2,\ldots,n$ such that, if on the circular table there are $n$ people seated and for all $k=1,2,\ldots,n$ the $k$ -th person is moving $i_n$ places in the right, all people will sit on different places.

V. Drenski

Let $f(x)$ and $g(x)$ be non-constant polynomials with integer positive coefficients, $m$ and $n$ are given natural numbers. Prove that there exists infinitely many natural numbers $k$ for which the numbers $$f(m^n)+g(0),f(m^n)+g(1),\ldots,f(m^n)+g(k)$$ are composite.

I. Tonov

(a) Find all real numbers $p$ for which the inequality $$x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3)$$ is true for all real numbers $x_1,x_2,x_3$ . (b) Find all real numbers $q$ for which the inequality $$x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4)$$ is true for all real numbers $x_1,x_2,x_3,x_4$ .

I. Tonov

Find the maximal count of shapes that can be placed over a chessboard with size $8\times8$ in such a way that no three shapes are not on two squares, lying next to each other by diagonal parallel $A1-H8$ ( $A1$ is the lowest-bottom left corner of the chessboard, $H8$ is the highest-upper right corner of the chessboard).

V. Chukanov

Find all point $M$ lying into given acute-angled triangle $ABC$ and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from $M$ to the lines $BC$ , $CA$ , $AB$ is maximal.

H. Lesov

In triangle pyramid $MABC$ at least two of the plane angles next to the edge $M$ are not equal to each other. Prove that if the bisectors of these angles form the same angle with the angle bisector of the third plane angle, the following inequality is true $$8a_1b_1c_1\le a^2a_1+b^2b_1+c^2c_1$$ where $a,b,c$ are sides of triangle $ABC$ and $a_1,b_1,c_1$ are edges crossed respectively with $a,b,c$ .

V. Petkov