The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

A large flat plate of glass is suspended $\sqrt{2/3}$ units above a large flat plate of wood. (The glass is infinitely thin and causes no funny refractive effects.) A point source of light is suspended $\sqrt{6}$ units above the glass plate. An object rests on the glass plate of the following description. Its base is an isosceles trapezoid $ABCD$ with $AB \parallel DC$, $AB = AD = BC = 1$, and $DC = 2$. The point source of light is directly above the midpoint of $CD$. The object’s upper face is a triangle $EF G$ with $EF = 2$, $EG = F G =\sqrt3$. $G$ and $AB$ lie on opposite sides of the rectangle $EF CD$. The other sides of the object are $EA = ED = 1$, $F B = F C = 1$, and $GD = GC = 2$. Compute the area of the shadow that the object casts on the wood plate.

Let $ a,b$ be two complex numbers. Prove that roots of $ z^{4}\plus{}az^{2}\plus{}b$ form a rhombus with origin as center, if and only if $ \frac{a^{2}}{b}$ is a non-positive real number.

Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$

We have a scalene acute triangle $ABC$ (triangle with no two equal sides) and a point $D$ on side $BC$. Pick a point $E$ on side $AB$ and a point $F$ on side $AC$ such that $\angle DEB=\angle DFC$. Lines $DF,\, DE$ intersect $AB,\, AC$ at points $M,\, N$, respectively. Denote $(I_1),\, (I_2)$ by the circumcircles of triangles $DEM,\, DFN$ in that order. The circle $(J_1)$ touches $(I_1)$ internally at $D$ and touches $AB$ at $K$, circle $(J_2)$ touches $(I_2)$ internally at $D$ and touches $AC$ at $H$. $P$ is the intersection of $(I_1),\, (I_2)$ different from $D$. $Q$ is the intersection of $(J_1),\, (J_2)$ different from $D$. a. Prove that all points $D,\, P,\, Q$ lie on the same line. b. The circumcircles of triangles $AEF,\, AHK$ intersect at $A,\, G$. $(AEF)$ also cut $AQ$ at $A,\, L$. Prove that the tangent at $D$ of $(DQG)$ cuts $EF$ at a point on $(DLG)$.

A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots. [i]Proposed by P. Kozhevnikov[/i]

Can four equal polygons be placed on the plane in such a way that any two of them don't have common interior points, but have a common boundary segment? (S.Markelov)

Given a point $O$ inside triangle $ABC$ . Prove that $$S_A * \overrightarrow{OA} + S_B * \overrightarrow{OB} + S_C * \overrightarrow{OC} = \overrightarrow{0}$$ where $S_A, S_B, S_C$ denote areas of triangles $BOC, COA, AOB$ respectively.

Bubble Boy and Bubble Girl live in bubbles of unit radii centered at $(20, 13)$ and $(0, 10)$ respectively. Because Bubble Boy loves Bubble Girl, he wants to reach her as quickly as possible, but he needs to bring a gift; luckily, there are plenty of gifts along the $x$-axis. Assuming that Bubble Girl remains stationary, find the length of the shortest path Bubble Boy can take to visit the $x$-axis and then reach Bubble Girl (the bubble is a solid boundary, and anything the bubble can touch, Bubble Boy can touch too)

Let $\omega$ be the circumscribed circle of the triangle $ABC$, in which $AC< AB$, $K$ is the center of the arc $BAC$, $KW$ is the diameter of the circle $\omega$. The circle $\gamma$ is inscribed in the curvilinear triangle formed by the segments $BC$, $AB$ and the arc $AC$ of the circle $\omega$. It turned out that circle $\gamma$ also touches $KW$ at point $F$. Let $I$ be the center of the triangle $ABC$, $M$ is the midpoint of the smaller arc $AK$, and $T$ is the second intersection point of $MI$ with the circle $\omega$. Prove that lines $FI$, $TW$ and $BC$ intersect at one point. (Mykhailo Sydorenko)

In the convex quadrilateral of the $ABCD$, the diagonals of $AC$ and $BD$ are mutually perpendicular and intersect at point $E$. On the side of $AD$, a point $P$ is chosen such that $PE = EC$. The circumscribed circle of the triangle $BCD$ intersects the segment $AD$ at the point $Q$. The circle passing through point $A$ and tangent to the line $EP$ at point $P$ intersects the segment $AC$ at point $R$. It turns out that points $B, Q, R$ are collinear. Prove that $\angle BCD = 90^o$.

Consider an acutangle triangle $ABC$ with circumcenter $O$. A circumference that passes through $B$ and $O$ intersect sides $BC$ and $AB$ in points $P$ and $Q$. Prove that the orthocenter of triangle $OPQ$ is on $AC$.

$E$ and $F$ are interior points of convex quadrilateral $ABCD$ such that $AE = BE$, $CE = DE$, $\angle AEB = \angle CED$, $AF = DF$, $BF = CF$, $\angle AFD = \angle BFC$. Prove that $\angle AFD + \angle AEB = \pi$.

Let $N, P$ be the centers of the faces A$BB'A'$ and $ADD'A'$, respectively, of a right parallelepiped $ABCDA'B'C'D'$ and $M \in (A'C)$ such that $A'M= \frac13 A' C$. Prove that $MN \perp AB'$ and $ MP \perp AD' $ if and only if the parallelepiped is a cube.

A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary. [i]Author: Kei Irie, Japan[/i]

Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$, choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen.

Consider a triangle $\vartriangle ACE$ with $\angle ACE = 45^o$ and $\angle CEA = 75^o$. Define points $Q, R$, and $P$ such that $AQ$, $CR$, and $EP$ are the altitudes of $\vartriangle ACE$. Let $H$ be the intersection of $AQ$, $CR$, and $EP$. Next define points $B, D$, and $F$ as follows. Extend $EP$ to point $B$ such that $BP = HP$, extend $AQ$ to point $D$ such that $DQ = HQ$, and extend $CR$ to point $F$ such that $F R = HR$. Finally, lengths $CH = 2$, $AH =\sqrt2$, and $EH =\sqrt3 - 1$. Compute the area of hexagon $ABCDEF$.

For a $\triangle ABC$ , let $A_1$ be the symmetric point of the intersection of angle bisector of $\angle BAC$ and $BC$ , where center of the symmetry is the midpoint of side $BC$, In the same way we define $B_1 $ ( on $AC$ ) and $C_1$ (on $AB$). Intersection of circumcircle of $\triangle A_1B_1C_1$ and line $AB$ is the set $\{Z,C_1 \}$, with $BC$ is the set $\{X,A_1\}$ and with $CA$ is the set $\{Y,B_1\}$. If the perpendicular lines from $X,Y,Z$ on $BC,CA$ and $ AB$ , respectively are concurrent , prove that $\triangle ABC$ is isosceles.

Given a triangle $ABC$. Consider all the tetrahedrons $PABC$ with $PH$ -- the smallest of all tetrahedron's heights. Describe the set of all possible points $H$.

Given $R, r > 0$. Two circles are drawn radius $R$, $r$ which meet in two points. The line joining the two points is a distance $D$ from the center of one circle and a distance $d$ from the center of the other. What is the smallest possible value for $D+d$?

Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$, respectively, are externally tangent at point $\lambda$. A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$, $O_2$ lie inside $C$. The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$. If $D$ is the midpoint of the segment $KL$, show that $\angle O_1OO_2 = \angle ADB$. [i]Greece[/i]

Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.

Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that $$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$

In an isosceles, obtuse-angled triangle, the lengths of two internal angle bisectors are in a $2:1$ ratio. Find the obtuse angle of the triangle.