Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$.

Given a square sheet of paper with side $1$. Measure on this sheet a distance of $ 5/6$. (The sheet can be folded, including, along any segment with ends at the edges of the paper and unbend back, after unfolding, a trace of the fold line remains on the paper).

Two circles $w_1$ and $w_2$ intersect at points $A$ and $B$. Tangents $\ell_1$ and $\ell_2$ respectively are drawn to them through point $A$. The perpendiculars dropped from point $B$ to $\ell_2$ and $\ell_1$ intersects the circles $w_1$ and $w_2$, respectively, at points $K$ and $N$. Prove that points $K, A$ and $N$ lie on one straight line.

An isosceles triangle $ABC$ with base $AC$ is given. Point $H$ is the intersection of altitudes. On the sides $AB$ and $BC$, points $M$ and $K$ are selected, respectively, so that the angle $KMH$ is right. Prove that a right-angled triangle can be constructed from the segments $AK, CM$ and $MK$.

Points $K$ and $M$ are taken on the sides $AB$ and $CD$ of square $ABCD$, respectively, and on the diagonal $AC$ - point $L$ such that $ML = KL$. Let $P$ be the intersection point of the segments $MK$ and $BD$. Find the angle $\angle KPL$.

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

Quadrangle $ABCD$ is inscribed in a circle. The perpendicular from the vertex $C$ on the bisector of $\angle ABD$ intersects the line $AB$ at the point $C_1$. The perpendicular from the vertex $B$ on the bisector of $\angle ACD$ intersects the line $CD$ at the point $B_1$. Prove that $B_1C_1 \parallel AD$.

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $K$ are taken, respectively, so that $S_{KMC} + S_{KAC}=S_{ABC}$. Prove that all such lines $MK$ pass through one point.

From the vertex $A$ of the parallelogram $ABCD$, the perpendiculars $AM,AN$ on sides $BC,CD$ respectively. $P$ is the intersection point of $BN$ and $DM$. Prove that the lines $AP$ and $MN$ are perpendicular.

All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.

In a triangle $ABC, O$ is the center of the circumscribed circle. Line $a$ passes through the midpoint of the altitude of the triangle from the vertex $A$ and is parallel to $OA$. Similarly, the straight lines $b$ and $c$ are defined. Prove that these three lines intersect at one point.