The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$. (A Zaslavsky)

Determine the ratio of the sides of the rectangle circumscribed around a corner of five cells (see figure). (M. Evdokimov)

On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$. (M. Volchkevich)

An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal. (M. Volchkevich)

Equilateral triangles $ABC_1, BCA_1, CAB_1$ are built on the sides of the triangle $ABC$ to the outside. On the segment $A_1B_1$ to the outer side of the triangle $A_1B_1C_1$, an equilateral triangle $A_1B_1C_2$ is constructed. Prove that $C$ is the midpoint of the segment $C_1C_2$. (A. Zaslavsky)

In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the intersection point of the medians of the triangle cuts off an equilateral triangle from it. (A. Zaslavsky)

An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles. (L. Blinkov)

Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron? (S. Markelov)

Two non-rolling circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ and radii $2R$ and $R$, respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on $C_1$ and the other two lie on $C_2$. (B. Frenkin)

The quadrangle $ABCD$ is inscribed in a circle, the center $O$ of which lies inside it. The tangents to the circle at points $A$ and $C$ and a straight line, symmetric to $BD$ wrt point $O$, intersect at one point. Prove that the products of the distances from $O$ to opposite sides of the quadrilateral are equal. (A. Zaslavsky)

The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle? (M. Volchkevich)

Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar. (A. Zaslavsky)