SHARYGIN 2007

A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?

Three circles pass through a point $P$ , and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the intersections of lines $AB_1$ and $BA_1$ . Let $A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,

Suppose two convex quadrangles are such that the sides of each of them lie on the perpendicular bisectors of the sides of the other one. Determine their angles,

Find the locus of centers of regular triangles such that three given points $A, B, C$ lie respectively on three lines containing sides of the triangle.

A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father's shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases? (Find the answer to $0,1$ , assuming that the radius of Earth equals $6000$ km.)

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$ . Prove that $PQ \perp AB$ .

On the side $AB$ of a triangle $ABC$ , two points $X, Y$ are chosen so that $AX = BY$ . Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$ . Prove that all lines $UV$ (for all $X, Y$ , given $A, B, C$ ) have a common point.

In a trapezium with bases $AD$ and $BC$ , let $P$ and $Q$ be the middles of diagonals $AC$ and $BD$ respectively. Prove that if $\angle DAQ = \angle CAB$ then $\angle PBA = \angle DBC$ .

In a triangle $ABC$ , let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$ . Prove that $\angle PAC = \angle QAB$ .

On two sides of an angle, points $A, B$ are chosen. The midpoint $M$ of the segment $AB$ belongs to two lines such that one of them meets the sides of the angle at points $A_1, B_1$ , and the other at points $A_2, B_2$ . The lines $A_1B_2$ and $A_2B_1$ meet $AB$ at points $P$ and $Q$ . Prove that $M$ is the midpoint of $PQ$ .

What triangles can be cut into three triangles having equal radii of circumcircles?

Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.

Into an angle $A$ of size $a$ , a circle is inscribed tangent to its sides at points $B$ and $C$ . A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

The base of a pyramid is a regular triangle having side of size $1$ . Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

There are two pipes on the plane (the pipes are circular cylinders of equal size, $4$ m around). Two of them are parallel and, being tangent one to another in the common generatrix, form a tunnel over the plane. The third pipe is perpendicular to two others and cuts out a chamber in the tunnel. Determine the area of the surface of this chamber.