Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$. [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

Let $ABCD$ be a parallelogram. On the ray $(DB$ a point $E$ is given such that the ray $(AB$ is the angle bisector of $\angle CAE$. Let $F$ be the intersection of $CE$ and $AB$. Prove that $\frac{AB}{BF} - \frac{AC}{AE} = 1$

Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Choose points $X,Y,Z$ on the segments $LA,LB,LC,$ respectively such that:$$\angle XBA=\angle YAB,\angle XCA=\angle ZAC.$$ Prove that: $\angle ZBC=\angle YCB.$

The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled? (S. Markelov)

$A,B,C$ are the three angles in a triangle such that $2\sin B\sin (A+B)-\cos A=1$, $2\sin C\sin (B+C)-\cos B=0$ find the three angles.

Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$) Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.

In the figure below, $\triangle ABC$ is an equilateral triangle. [asy] import graph; unitsize(60); axes("$x$", "$y$", (0, 0), (1.5, 1.5), EndArrow); real w = sqrt(3) - 1; pair A = (1, 1); pair B = (0, w); pair C = (w, 0); draw(A -- B -- C -- cycle); dot(Label("$A(1, 1)$", A, NE), A); dot(Label("$B$", B, W), B); dot(Label("$C$", C, S), C); [/asy] Point $A$ has coordinates $(1, 1)$, point $B$ is on the positive $y$-axis, and point $C$ is on the positive $x$-axis. What is the area of $\triangle ABC$?

a) Given a triangle $ABC$ with $\angle BAC=90^{\circ}$ and the altitude $AH(H$ is on the side $BC).$ Prove that: the [i]Lemoine[/i] point of the triangle $ABC$ is the midpoint of $AH.$ b) If a triangle has its [i]Lemoine[/i] point is the midpoint of $1$ in $3$ symmedian segments, does that triangle need to be a right triangle? Explain why.

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach? $ \text{(A)}\ 2\pi/3 \qquad \text{(B)}\ 2\pi \qquad \text{(C)}\ 5\pi/2 \qquad \text{(D)}\ 8\pi/3 \qquad \text{(E)}\ 3\pi$

Consider the parabola $ P:x-y^2-(p+3)y-p=0,p\in\mathbb{R}^*. $ Show that $ P $ intersects the coordonate axis at three points, and that the circle formed by these three points passes through a fixed point.

The perimeter of a triangle is a natural number, its circumradius is equal to $\frac{65}{8}$, and the inradius is equal to $4$. Find the sides of the triangle.

Side $BC$ of triangle $ABC$ is equal to $a$ and the opposite angle is equal to $\theta$. A straight line passing through the midpoint of $BC$ and the center of the inscribed circle intersects lines $AB$ and $AC$ at points $M$ and $P$, respectively. Find the area of the quadrilateral (non-convex) $BMPC$.

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

We have an equilateral triangle with circumradius $1$. We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$, is maximum. (The total distance of the point P from the sides of an equilateral triangle is fixed ) [i]Proposed by Erfan Salavati[/i]

Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.

Let $ABC$ be a acute angled triangle and let $AD,BE,CF$ be its altitudes. $X \not=A,B$ and $Y \not=A,C$ lie on sides $AB$ and $AC$, respectively, so that $ADXY$ is a cyclic quadrilateral. Let $H$ be the orthocenter of triangle $AXY$. Prove that $H$ lies on line $EF$.

Two vertices $A,B$ of a triangle $ABC$ are located on a parabola $y=ax^2 + bx + c$ with $a>0$ in such a way that the sides $AC,BC$ are tangent to the parabola. Let $m_c$ be the length of the median $CC_1$ of triangle $ABC$ and $S$ be the area of triangle $ABC$. Find \[\frac{S^2}{m_c^3}\]

Let $ABCD$ be a quadrilateral such that $\angle A$ and $\angle D$ are acute and $\overline{AB} = \overline{BC} = \overline{CD}$. Suppose that $\angle BDA = 30^\circ$, prove that $\angle DAC= 30^\circ$.

Let $ f$ be an entire function on $ \mathbb C$ and $ \omega_1,\omega_2$ are complex numbers such that $ \frac {\omega_1}{\omega_2}\in{\mathbb C}\backslash{\mathbb Q}$. Prove that if for each $ z\in \mathbb C$, $ f(z) \equal{} f(z \plus{} \omega_1) \equal{} f(z \plus{} \omega_2)$ then $ f$ is constant.

We call a bar of width ${w}$ on the surface of the unit sphere ${\Bbb{S}^2}$, a spherical segment, centered at the origin, which has width ${w}$ and is symmetric with respect to the origin. Prove that there exists a constant ${c>0}$, such that for any positive integer ${n}$ the surface ${\Bbb{S}^2}$ can be covered with ${n}$ bars of the same width so that any point is contained in no more than ${c\sqrt{n}}$ bars.

A circle is drawn through vertices $A$ and $B$ of triangle $ABC$, intersecting sides $AC$ and $BC$ at points $M$ and $P$. It is known that the segment $MP$ contains the center of the circle inscribed in $ABC$. Find $MP$ if $AB = c$, $BC = a$, $CA=b$.

In $\triangle ABC$, $\angle ACB=3\angle BAC$, $BC=5$, $AB=11$. Find $AC$

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

There are $n$ non-coplanar points in space. Prove that there exists a circle exactly passes through three points of them.

Let $ABC$ be a triangle with incenter $I$ and circumcircle $\Gamma$. Circles $\omega_{B}$ passing through $B$ and $\omega_{C}$ passing through $C$ are tangent at $I$. Let $\omega_{B}$ meet minor arc $AB$ of $\Gamma$ at $P$ and $AB$ at $M\neq B$, and let $\omega_{C}$ meet minor arc $AC$ of $\Gamma$ at $Q$ and $AC$ at $N\neq C$. Rays $PM$ and $QN$ meet at $X$. Let $Y$ be a point such that $YB$ is tangent to $\omega_{B}$ and $YC$ is tangent to $\omega_{C}$. Show that $A,X,Y$ are collinear.