SHARYGIN 2011

Let $H$ be the orthocenter of triangle $ABC$ . The tangents to the circumcircles of triangles $CHB$ and $AHB$ at point $H$ meet $AC$ at points $A_1$ and $C_1$ respectively. Prove that $A_1H = C_1H$ .

The diagonals of trapezoid $ABCD$ meet at point $O$ . Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$ .

The excircle of right-angled triangle $ABC$ ( $\angle B =90^o$ ) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$ . Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$ , point $C'$ is defined similarly. Prove that $AC||A'C'$ .

Let $AP$ and $BQ$ be the altitudes of acute-angled triangle $ABC$ . Using a compass and a ruler, construct a point $M$ on side $AB$ such that $\angle AQM = \angle BPM$ .

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex). b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

In triangle $ABC$ , the altitude and the median from vertex $A$ form (together with line $BC$ ) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$ .

Given a circle with center $O$ and radius equal to $1$ . $AB$ and $AC$ are the tangents to this circle from point $A$ . Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$ .

Given are triangle $ABC$ and line $\ell$ . The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$ . Points $B_1, C_1$ are defined similarly. Prove that a) lines $AA_1, BB_1, CC_1$ concur, b) their common point lies on the circumcircle of $ABC$ c) two points constructed in this way for two perpendicular lines are opposite.

a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix? b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

On the plane, given are $n$ lines in general position, i.e. any two of them aren't parallel and any three of them don't concur. These lines divide the plane into several parts. What is a) the minimal, b) the maximal number of these parts that can be angles?

Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?

Quadrilateral $ABCD$ is circumscribed around a circle with center $I$ . Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$ . Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$ .

Nikolai Beluhov and Aleksey Zaslavsky

On a circle with diameter $AC$ , let $B$ be an arbitrary point distinct from $A$ and $C$ . Points $M, N$ are the midpoints of chords $AB, BC$ , and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$ , and lines $CP$ and $AB$ meet at point $L$ . Prove that lines $MQ, NP$ and $KL$ concur.

Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.

Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$ . Points $B'$ and $C'$ are defined similarly. (a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$ . (b) Suppose $\ell$ passes through the circumcenter of $\triangle ABC$ . Prove that in this case $\ell'$ passes through the center of its nine-points circle.

M. Marinov and N. Beluhov

Given is an acute-angled triangle $ABC$ . On sides $BC, CA, AB$ , find points $A', B', C'$ such that the longest side of triangle $A'B'C'$ is minimal.

Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?