BG-MO 1992

Through a random point $C_1$ from the edge $DC$ of the regular tetrahedron $ABCD$ is drawn a plane, parallel to the plane $ABC$ . The plane constructed intersects the edges $DA$ and $DB$ at the points $A_1,B_1$ respectively. Let the point $H$ is the midpoint of the altitude through the vertex $D$ of the tetrahedron $DA_1B_1C_1$ and $M$ is the center of gravity (barycenter) of the triangle $ABC_1$ . Prove that the measure of the angle $HMC$ doesn't depend on the position of the point $C_1$ . (Ivan Tonov)

Prove that there exists $1904$ -element subset of the set $\{1,2,\ldots,1992\}$ , which doesn't contain an arithmetic progression consisting of $41$ terms. (Ivan Tonov)

Let $m$ and $n$ are fixed natural numbers and $Oxy$ is a coordinate system in the plane. Find the total count of all possible situations of $n+m-1$ points $P_1(x_1,y_1),P_2(x_2,y_2),\ldots,P_{n+m-1}(x_{n+m-1},y_{n+m-1})$ in the plane for which the following conditions are satisfied:

(i) The numbers $x_i$ and $y_i~(i=1,2,\ldots,n+m-1)$ are integers and $1\le x_i\le n,1\le y_i\le m$ . (ii) Every one of the numbers $1,2,\ldots,n$ can be found in the sequence $x_1,x_2,\ldots,x_{n+m-1}$ and every one of the numbers $1,2,\ldots,m$ can be found in the sequence $y_1,y_2,\ldots,y_{n+m-1}$ . (iii) For every $i=1,2,\ldots,n+m-2$ the line $P_iP_{i+1}$ is parallel to one of the coordinate axes. (Ivan Gochev, Hristo Minchev)

Let $p$ be a prime number in the form $p=4k+3$ . Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$ , then at least one of them is divisible by $p$ . (Plamen Koshlukov)

Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$ . Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$ , and the line $MN$ intersects the line $CD$ at the point $O$ . Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$ . Prove that $CD=PQ$ . (Plamen Koshlukov)

There are given one black box and $n$ white boxes ( $n$ is a random natural number). White boxes are numbered with the numbers $1,2,\ldots,n$ . In them are put $n$ balls. It is allowed the following rearrangement of the balls: if in the box with number $k$ there are exactly $k$ balls, that box is made empty - one of the balls is put in the black box and the other $k-1$ balls are put in the boxes with numbers: $1,2,\ldots,k-1$ . (Ivan Tonov)