SHARYGIN 2015

Let $ABC$ be an acute-angled triangle. Construct points $A', B', C'$ on its sides $BC, CA, AB$ such that:

• $A'B' \parallel AB$ ,
• $C'C$ is the bisector of angle $A'C'B'$ ,
• $A'C' + B'C'= AB$ .

The diagonals of a convex quadrilateral divide it into four similar triangles. Prove that is possible to inscribe a circle into this quadrilateral

Let $H$ be the orthocenter of an acute-angled triangle A $BC$ . The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ( $O$ is the circumcenter of triangle $ABC$ ) is equal to $AC$ .

Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$ . Point $M$ is the midpoint of $H_2H_3$ . Line $AM$ meets $H_2H_1$ at point $K$ . Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$ .

Let $ABC$ be an acute-angled, nonisosceles triangle. Point $A_1, A_2$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $BC$ . Segment $A_1A_2$ is a diameter of a circle $\alpha$ . Circles $\beta$  and $\gamma$ are defined similarly. Prove that these three circles have two common points.

The sidelengths of a triangle $ABC$ are not greater than $1$ . Prove that $p(1 -2Rr)$ is not greater than $1$ , where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$ .

The diagonals of a convex quadrilateral divide it into four triangles. Restore the quadrilateral by the circumcenters of two adjacent triangles and the incenters of two mutually opposite triangles

Let $O$ be the circumcenter of a triangle $ABC$ . The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $\ell$ and $L$ such that $\ell // XO$ . Prove that the angles formed by $L$ and by the diagonals of quadrilateral $ABCD$ are equal.

Let $ABCDEF$ be a cyclic hexagon, points $K, L, M, N$ be the common points of lines $AB$ and $CD$ , $AC$ and $BD$ , $AF$ and $DE$ , $AE$ and $DF$ respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$ . Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$ . The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$ . Prove that $Q$ lies on the medial line of triangle $LKP$ .

Given are a circle and an ellipse lying inside it with focus $C$ . Find the locus of the circumcenters of triangles $ABC$ , where $AB$ is a chord of the circle touching the ellipse.

A quadrilateral $ABCD$ is inscribed into a circle $\omega$ with center $O$ . Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\Omega$ be the circumcircle of triangle $OM_1M_2$ . Let $X_1$ and $X_2$ be the common points of $\omega$ and $\Omega$ and $Y_1$ and $Y_2$ the second common points of $\Omega$ with the circumcircles of triangles $CDM_1$ and $ABM_2$ . Prove that $X_1X_2 // Y_1Y_2$ .

The faces of an icosahedron are painted into $5$ colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.

A tetrahedron $ABCD$ is given. The incircles of triangles $ABC$ and $ABD$ with centers $O_1, O_2$ , touch $AB$ at points $T_1, T_2$ . The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$ . The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.

The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and $D_1$ . a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$ . Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $P$ . b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I$ with plane $B_1C_1D_1$ . Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$ .