Convex quadrilateral $ABCD$ is given. Lines $BC$ and $AD$ intersect at point $O$, with $B$ lying on the segment $OC$, and $A$ on the segment $OD$. $I$ is the center of the circle inscribed in the $OAB$ triangle, $J$ is the center of the circle exscribed in the triangle $OCD$ touching the side of $CD$ and the extensions of the other two sides. The perpendicular from the midpoint of the segment $IJ$ on the lines $BC$ and $AD$ intersect the corresponding sides of the quadrilateral (not the extension) at points $X$ and $Y$. Prove that the segment $XY$ divides the perimeter of the quadrilateral$ABCD$ in half, and from all segments with this property and ends on $BC$ and $AD$, segment $XY$ has the smallest length.

A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.

Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.

Two equal circles intersect at points $A$ and $B$. $P$ is the point of one of the circles that is different from $A$ and $B, X$ and $Y$ are the second intersection points of the lines of $PA, PB$ with the other circle. Prove that the line passing through $P$ and perpendicular to $AB$ divides one of the arcs $XY$ in half.

Construct a point $Q$ in triangle $ABC$ such that at least two of the segments $CQ, BQ, AQ$, divide the inscribed circle in half. For which triangles is this possible?

Points $K, L$ on sides $AC, CB$ respectively of a triangle $ABC$ are the points of contact of the excircles with the corresponding sides . Prove that the straight line through the midpoints of $KL$ and $AB$ (a) divides the perimeter of triangle $ABC$ in half, (b) is parallel to the bisector of angle $ACB$. ( L Emelianov)

In triangle $ABC$, the altitude $AH$ passes through midpoint of the median $BM$. Prove that in the triangle $BMC$ also one of the altitudes passes through the midpoint of one of the medians.

Let $ABC$ be an acute triangle with $AC>CB$ and let $M$ be the midpoint of side $AB$. Denote by $Q$ the midpoint of the big arc $AB$ which cointais $C$ and by $B_1$ the point inside $AC$ such that $BC=CB_1$. $B_1Q$ touches $BC$ in $E$ and $K$ is the intersection of $(BB_1M)$ and $(ABC)$. Prove that $KC$ bissects $B_1E$.

Prove that the plane dividing in equal proportions the surface area and volume of the circumscribed polyhedron passes through the center of the sphere inscribed in this polyhedron.

A circle can be drawn around the quadrilateral $ABCD$. Let straight lines $AB$ and $CD$ intersect at point $M$, and straight lines $BC$ and $AD$ intersect at point $K$. (Points $B$ and $P$ lie on segments $AM$ and $AK$, respectively.) Let $P$ be the projection of point $M$ onto straight line $AK$, $L$ be the projection of point $K$ on the straight line $AM$. Prove that the straight line $LP$ divides the diagonal $BD$ in half.

Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.