In an isosceles trapezoid, the diagonals are perpendicular. Find the distance from the center of the circle described around the trapezoid to the point of intersection of its diagonals, if the lengths of the bases are equal to $a$ and $b$.

The bisector $BL$ was drawn in the triangle $ABC$. Let the points $I_1$ and $I_2$ be centers of the circles inscribed in the triangles $ABL$ and $CBL$, and the points $J_1$ and $J_2$ be centers of the excircles of these triangles touching the side $BL$. Prove that the points $I_1$, $I_2$, $J_1$ and $J_2$ lie on the same circle.

An equilateral triangle $ABE$ is built inside the square $ABCD$ on the side $AB$, and an equilateral triangle $AFC$ is built on the diagonal $AC$ ($D$ is inside this triangle). The segment $EF$ intersects $CD$ at point $P$. Prove that the lines $AP$, $BE$ and $CF$ intersect at the same point.

Given a triangle $ABC$ in which the angle $B$ is equal to $60^\circ$. A circle inscribed in a triangle with a center $I$ touches the side $AC$ at point $K$. A line passing through the points of touching of this circle with the other sides of the triangle intersects the its circumcircle at points $M$ and $N$. Prove that the ray $KI$ divides the arc $MN$ in half.

An acute-angled unequal triangle $ABC$ is drawn with its circumcircle and circumcenter $O$. The incenter $I$ is also marked. Using only a ruler (without divisions), construct the symedian (a line symmetrical to the median relative to the corresponding bisector) of the triangle, drawing no more than four lines.

Given an acute-angled triangle $ABC$ and a point $P$ inside it such that $\angle PBA=\angle PCA$. The lines $PB$ and $PC$ intersect the circumcircles of triangles $PCA$ and $PAB$ secondly at points $M$ and $N$, respectively. Let the rays $MC$ and $NB$ intersect at a point $S$, $K$ is the center of the circumscribed circle of the triangle $SMN$. Prove that the lines $AK$ and $BC$ are perpendicular.

In a plane: 1. An ellipse with foci $F_1$, $F_2$ lies inside a circle $\omega$. Construct a chord $AB$ of $\omega$. touching the ellipse and such that $A$, $B$, $F_1$, and $F_2$ are concyclic. 2. Let a point $P$ lie inside an acute angled triangle $ABC$, and $A'$, $B'$, $C'$ be the projections of $P$ to $BC$, $CA$, $AB$ respectively. Prove that the diameter of circle $A'B'C'$ equals $CP$ if and only if the circle $ABP$ passes through the circumcenter of $ABC$. Proposed by Alexey Zaslavsky

Petya drew a pentagon $ABCDE$ on the plane. After that, Vasya marked all the points $S$ in a given half-space relative to the plane of the pentagon so that in the pyramid $SABCD$ exactly two side faces are perpendicular to the plane of the base $ABCD$, and the height is $1$. How many points could have Vasya?

The hypotenuse $AB$ of a right-angled triangle $ABC$ touches the corresponding excircle $\omega$ at point $T$. Point $S$ is symmetrical $T$ relative to the bisector of angle $C$, $CH$ is the height of the triangle. Prove that the circumcircle of triangle $CSH$ touches the circle $\omega$.

Straight lines are drawn containing the sides of an unequal triangle $ABC$, its incircle $I$ circle and a its circumcircle, the center of which is not marked. Using only a ruler (without divisions), construct the symedian of the triangle (a straight line symmetrical to the median relative to the corresponding bisector), drawing no more than six lines.

From point $D$ of parallelogram $ABCD$ were drawn an arbitrary line $\ell_1$, intersecting the segment $AB$ and the line $BC$ at points $C_1$ and $A_1$, respectively, and an arbitrary line $\ell_2$ intersecting the segment $BC$ and the line $AB$ at the points $A_2$ and $C_2$, respectively. Find the locus of the intersection points of the circles $(A_1BC_2)$ and $(A_2BC_1)$ (other than point $B$).

An unequal acute-angled triangle $ABC$ with an orthocenter $H$ is given, $M$ is the midpoint of side $BC$. Points $K$ and $L$ lie on a line passing through $H$ and perpendicular to $AM$ such a $KB$ and $LC$ perpendicular to $BC$. Point $N$ lies on the line $HM$, and the lines $AN$ and $AH$ are symmetric with respect to the line $AM$. Prove that a circle with a diameter $AN$ touches two circles: centered at $K$ and with a radius $KB$ and with a center $L$ and radius $LC$.