The hexagon has five $90^o$ angles and one $270^o$ angle (see picture). Use a straight-line ruler to divide it into two equal-sized polygons.

A parallelogram of $ABCD$ is given. Line parallel to $AB$ intersects the bisectors of angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the angles $ADP$ and $ABQ$ are equal. (A. Hakobyan)

In triangle $ABC$, points $K ,P$ are chosen on the side $AB$ so that $AK = BL$, and points $M,N$ are chosen on the side $BC$ so that $CN = BM$. Prove that $KN + LM \ge AC$. (I. Bogdanov)

Given a hexagon $ABCDEF$, in which $AB = BC, CD = DE, EF = FA$, and angles $A$ and $C$ are right. Prove that lines $FD$ and $BE$ are perpendicular. (B. Kukushkin)

The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle. (A. Zaslavsky)

Let $A_1,B_1,C_1$ are the midpoints of the sides of the triangle $ABC, I$ is the center of the circle inscribed in it. Let $C_2$ be the intersection point of lines $C_1 I$ and $A_1B_1$. Let $C_3$ be the intersection point of lines $CC_2$ and $AB$. Prove that line $IC_3$ is perpendicular to line $AB$. (A. Zaslavsky)

Given an acute-angled triangle $ABC$. A straight line parallel to $BC$ intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At what location of the points $M$ and $P$ will the radius of the circle circumscribed about the triangle $BMP$ be the smallest? (I. Sharygin)

On a circle with diameter $AB$, lie points $C$ and $D$. $XY$ is the diameter passing through the midpoint $K$ of the chord $CD$. Point $M$ is the projection of point $X$ onto line $AC$, and point $N$ is the projection of point $Y$ on line $BD$. Prove that points $M, N$ and $K$ are collinear. (A. Zaslavsky)

$ABCBE$ is a regular pentagon. Point $B'$ is symmetric to point $B$ wrt line $AC$ (see figure). Is it possible to pave the plane with pentagons equal to $AB'CBE$? (S. Markelov)

A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal. (M. Volchkevich)

An arbitrary point $M$ is chosen inside the triangle $ABC$. Prove that $MA + MB + MC \le max (AB + BC, BC + AC, AC + AB)$. (N. Sedrakyan)

Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle? (I. Bogdanov)