BG-MO 1963

Find all three-digit numbers whose remainders after division by $11$ give quotient, equal to the sum of the squares of its digits.

In the trapezium $ABCD$ , a point $M$ is chosen on the non-base segment $AB$ . Through the points $M,A,D$ and $M,B,C$ are drawn circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ . Prove that:

(a) the second intersection point $N$ of $k_1$ and $k_2$ lies on the other non-base segment $CD$ or on its continuation; (b) the length of the line $O_1O_2$ doesn't depend on the location of $M$ on $AB$ ; (c) the triangles $O_1MO_2$ and $DMC$ are similar. Find such a position of $M$ on $AB$ that makes $k_1$ and $k_2$ have the same radius.

In the tetrahedron $ABCD$ three of the faces are right-angled triangles and the other is not an obtuse triangle. Prove that:

(a) the fourth wall of the tetrahedron is a right-angled triangle if and only if exactly two of the plane angles having common vertex with the some of vertices of the tetrahedron are equal. (b) its volume is equal to $\frac16$ multiplied by the multiple of two shortest edges and an edge not lying on the same wall.