SHARYGIN 2009

Given $n$ points on the plane, which are the vertices of a convex polygon, $n > 3$ . There exists $k$ regular triangles with the side equal to $1$ and the vertices at the given points.

• Prove that $k < \frac {2}{3}n$ .
• Construct the configuration with $k > 0.666n$ .

Let $ABC$ be an acute triangle, $CC_1$ its bisector, $O$ its circumcenter. The perpendicular from $C$ to $AB$ meets line $OC_1$ in a point lying on the circumcircle of $AOB$ . Determine angle $C$ .

Given quadrilateral $ABCD$ . The circumcircle of $ABC$ is tangent to side $CD$ , and the circumcircle of $ACD$ is tangent to side $AB$ . Prove that the length of diagonal $AC$ is less than the distance between the midpoints of $AB$ and $CD$ .

Let $CL$ be a bisector of triangle $ABC$ . Points $A_1$ and $B_1$ are the reflections of $A$ and $B$ in $CL$ , points $A_2$ and $B_2$ are the reflections of $A$ and $B$ in $L$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $AB_1B_2$ and $BA_1A_2$ respectively. Prove that angles $O_1CA$ and $O_2CB$ are equal.

In triangle $ABC$ , one has marked the incenter, the foot of altitude from vertex $C$ and the center of the excircle tangent to side $AB$ . After this, the triangle was erased. Restore it.

Given triangle $ABC$ of area 1. Let $BM$ be the perpendicular from $B$ to the bisector of angle $C$ . Determine the area of triangle $AMC$ .

Given a circle and a point $C$ not lying on this circle. Consider all triangles $ABC$ such that points $A$ and $B$ lie on the given circle. Prove that the triangle of maximal area is isosceles.

Three lines passing through point $O$ form equal angles by pairs. Points $A_1$ , $A_2$ on the first line and $B_1$ , $B_2$ on the second line are such that the common point $C_1$ of $A_1B_1$ and $A_2B_2$ lies on the third line. Let $C_2$ be the common point of $A_1B_2$ and $A_2B_1$ . Prove that angle $C_1OC_2$ is right.

Given triangle $ABC$ and two points $X$ , $Y$ not lying on its circumcircle. Let $A_1$ , $B_1$ , $C_1$ be the projections of $X$ to $BC$ , $CA$ , $AB$ , and $A_2$ , $B_2$ , $C_2$ be the projections of $Y$ . Prove that the perpendiculars from $A_1$ , $B_1$ , $C_1$ to $B_2C_2$ , $C_2A_2$ , $A_2B_2$ , respectively, concur if and only if line $XY$ passes through the circumcenter of $ABC$ .

Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).

Given convex $n$ -gon $A_1\ldots A_n$ . Let $P_i$ ( $i = 1,\ldots , n$ ) be such points on its boundary that $A_iP_i$ bisects the area of polygon. All points $P_i$ don't coincide with any vertex and lie on $k$ sides of $n$ -gon. What is the maximal and the minimal value of $k$ for each given $n$ ?

Suppose $H$ and $O$ are the orthocenter and the circumcenter of acute triangle $ABC$ ; $AA_1$ , $BB_1$ and $CC_1$ are the altitudes of the triangle. Point $C_2$ is the reflection of $C$ in $A_1B_1$ . Prove that $H$ , $O$ , $C_1$ and $C_2$ are concyclic.

The opposite sidelines of quadrilateral $ABCD$ intersect at points $P$ and $Q$ . Two lines passing through these points meet the side of $ABCD$ in four points which are the vertices of a parallelogram. Prove that the center of this parallelogram lies on the line passing through the midpoints of diagonals of $ABCD$ .

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

Is it true that for each $n$ , the regular $2n$ -gon is a projection of some polyhedron having not greater than $n + 2$ faces?

A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic.