Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$.

A trapezoid is given in which one base is twice as large as the other. Use one ruler (no divisions) to draw the midline of this trapezoid.

$ABCD$ is a convex quadrilateral such that $\angle A = \angle C < 90^{\circ}$ and $\angle ABD = 90^{\circ}$. $M$ is the midpoint of $AC$. Prove that $MB$ is perpendicular to $CD$.

On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.

Point $M$ is a midpoint of side $BC$ of a triangle $ABC$ and $H$ is the orthocenter of $ABC$. $MH$ intersects the $A$-angle bisector at $Q$. Points $X$ and $Y$ are the projections of $Q$ on sides $AB$ and $AC$. Prove that $XY$ passes through $H$.

Quadrilateral $ABCD$ is inscribed in a circle, $E$ is an arbitrary point of this circle. It is known that distances from point $E$ to lines $AB, AC, BD$ and $CD$ are equal to $a, b, c$ and $d$ respectively. Prove that $ad= bc$.

Two quadrangles have equal areas, perimeters and corresponding angles. Are such quadrilaterals necessarily congurent ?

Circle $(O)$ and its chord $BC$ are given. Point $A$ moves on the major arc $BC$. $AL$ is the angle bisector in a triangle $ABC$. Show that the disctance from the circumcenter of triangle $AOL$ to the line $BC$ does not depend on the position of point $A$.

Points $STABCD$ in space form a convex octahedron with faces $SAB,SBC,SCD,SDA,TAB,TBC,TCD,TDA$ such that there exists a sphere that is tangent to all of its edges. Prove that $A,B,C,D$ lie in one plane.

Let $ABC$ be a triangle, $I$ and $O$ be its incenter and circumcenter respectively. $A'$ is symmetric to $O$ with respect to line $AI$. Points $B'$ and $C'$ are defined similarly. Prove that the nine-point centers of triangles $ABC$ and $A'B'C'$ coincide.

$ABCD$ is a square and $XYZ$ is an equilateral triangle such that $X$ lies on $AB$, $Y$ lies on $BC$ and $Z$ lies on $DA$. Line through the centers of $ABCD$ and $XYZ$ intersects $CD$ at $T$. Find angle $CTY$