Sel-Geo.ru

 Selected geometry problems from geometry.ru from various proposers (from here)

collected inside aops here

Nikolai Beluhov

1. The point $F$ lies inside $ABC$ and is such that $\angle AFB = \angle BFC = \angle CFA = 120^o$ . Let $A_1 = AF  \cap  BC$, $B_1 = BF  \cap  CA$, $C_1 = CF \cap AB$. Show that the Euler lines of the triangles $AFB_1, BFC_1, CFA_1$ form an equilateral triangle of perimeter $AA_1 + BB_1 + CC_1$.

[Matematika+, 2006]

2. Let $ABCDEF$ be a non-convex hexagon which has no parallel sides and in which $AB = DE$, $BC = EF$, $CD = FA$, $\angle FAB = 3\angle  CDE$, $\angle BCD = 3\angle EFA$, $\angle DEF = 3\angle ABC$. Show that the lines $AD, BE, CF$ are concurrent. 

[Matematika, 2009 -  Kvant, 2009]

3. The two circles $\omega_1$ and $\omega_2$ meet in $A$ and $B$ and their common external tangents meet in $O$. The line $\ell$ through $O$ meets $\omega_1$ and $\omega_2$ in the points $P$ and $Q$ closer to $O$. Let $M = AP \cap BQ$, $N = AQ \cap BP$, and let $C \in\ell$ be such that $CM = CN = \alpha$. Show that $\alpha$ remains constant when $\ell$ varies.

 [Matematika, 2009]

4. The diagonals $AC, BD$ of the convex quadrilateral $ABCD$ meet at $O$ and the bisectors (internal & external) of $\angle AOB$ meet the segments $\overline{AB},\overline{BC},\overline{CD},\overline{DA}$ at $M, N, P, Q$ respectively. Prove that the medians from $A$ in $\triangle AMQ$, from $B$ in $\triangle BMN$, from $C$ in $\triangle CNP$ and from $D$ in $\triangle DPQ$ are concurrent.

[Unpublished, 2009]

5. Let $ABCD$ be a circumscribed quadrilateral. Let $E = AC\cap BD$ and let $I_a, I_b, I_c, I_d$ be the incenters of $\vartriangle BCD, \vartriangle CDA, \vartriangle DAB, \vartriangle ABC$, respectively. Show that the segments $I_aI_c$ and $I_bI_d$ meet in the center of a circle which passes through the incenters of $\vartriangle AEB, \vartriangle BEC, \vartriangle CED, \vartriangle DEA$. 

 [Kvant, 2010]

6. Does there exist a linear function $f$ of five variables such that, for any triangle $ABC$ with circumradius $R$, inradius $r$, and exradii $r_a, r_b, r_c$ we have $f(R, r, r_a, r_b, r_c) = 0$ ?

[Unpublished, 2009]

Blinkov Alexander Davidovich

1. Is there a tetrahedron, all of whose faces are congruent right-angled triangles? 

[MMO]

2. Line segment $AB$ is a common chord of two circles with equal radius. Through an arbitrary point lying inside this segment, drawn to it perpendicular that intersects the circles at points $C$ and $D$ (in one of the half-planes with boundary $AB$). Prove that point $D$ is an orthocenter of  triangle $ABC$.

[Savin Tournament ]

 Blinkov Yuri Alexandrovich

1. The tangents of the cicrumcircle of triangle $ABC$ are drawn at points $A$ and $C$, intersecting at point $P$. Let $AA_1, BB_1$ and $CC_1$ are the altitudes of triangle $ABC$. Line $PB_1$ meets $A_1C_1$ at point $K$. Prove that the midpoint of the side $AC$, the orthocenter, and the point $K$ are collinear.

[Savin Tournament ]

2. In a right triangle $ABC$ with right angle $C$, angle $A$ is $30^o$.  Point $I$ is the center of the inscribed circle of the triangle $ABC, D$ is the intersection point of segment $BI$ with this circle. Prove that the segments $AI$ and $CD$  are perpendicular. 

[Moscow MMO 2011]

Kozhevnikov Pavel Alexandrovich

1. Points $K, L, M, N$  are taken on the edges $AB, BC, CD, DA$ of the tetrahedron $ABCD$ respectively. Points $K', L', M', N'$ are symmetric to the points $K, L, M, N$ wrt  the midpoints of edges $AB, BC, CD, DA$, respectively. Prove that the volumes of tetrahedra $KLMN$ and $K'L'M'N'$ are equal.

2. Circles $\omega_b$, $\omega_c$ are excircles for triangle $ABC$ (i.e. $\omega_b$ and $\omega_c$ touch, respectively, sides $AC$ and $AB$ and extensions of two other sides). Circle $\omega'_b$ is symmetric to $\omega_b$ wrt the midpoint of side $AC$, circle $\omega'_c$ is symmetric to $\omega_c$ wrt the midpoint of side $AB$. Prove that the line passing through the intersection points of the circles $\omega'_b$ and $\omega'_c$ , divides the perimeter of triangle $ABC$ in half.

3. The quadrilateral $ABCD$ is inscribed in the circle $\omega$, and its diagonals intersect at the point $K$. Points $M_1, M_2, M_3, M_4$ are the midpoints of arcs $AB, BC, CD, DA$ (not containing other vertices quadrangle), respectively. Points $I_1, I_2, I_3, I_4$ are the centers of the circles inscribed in triangles $ABK, BCK, CDK, DAK$ respectively. Prove that lines $M_1I_1, M_2I_2 , M_3I_3, M_4I_4$ intersect at one point.