Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.

In the plane let two circles be given which intersect at two points $A, B$; Let $PT$ be one of the two common tangent line of these circles ($P, T$ are points of tangency). Tangents at $P$ and $T$ of the circumcircle of triangle $APT$ meet each other at $S$. Let $H$ be a point symmetric to $B$ under $PT$. Show that $A, S, H$ are collinear.

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$. Let $M$ be a point on the side $BC$ such that $\angle BAM = \angle DAC$. Further, let $L$ be the second intersection point of the circumcircle of the triangle $CAM$ with the side $AB$, and let $K$ be the second intersection point of the circumcircle of the triangle $BAM$ with the side $AC$. Prove that $KL \parallel BC$.

Let $ ABC$ be a triangle and the points $ D\in (BC)$, $ E\in (CA)$, $ F\in (AB)$ such that \[ \frac {BD}{DC} \equal{} \frac {CE}{EA} \equal{} \frac {AF}{FB}.\] Prove that if the circumcenters of the triangles $ DEF$ and $ ABC$ coincide then $ ABC$ is equilateral.

Let $A'$ be the result of reflection of vertex $A$ of triangle ABC through line $BC$ and let $B'$ be the result of reflection of vertex $B$ through line $AC$. Given that $\angle BA' C = \angle BB'C$, can the largest angle of triangle $ABC$ be located: a) At vertex $A$, b) At vertex $B$, c) At vertex $C$?

Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$. Such an arrangement is called [i]successful[/i] if each number is the product of its neighbors. Find the number of successful arrangements.

Triangle $ABC$ has a right angle at $B$. Point $D$ lies on side $BC$ such that $3\angle BAD = \angle BAC$. Given $AC=2$ and $CD=1$, compute $BD$.

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]

The inscribed circle of triangle $ABC$ is tangent to sides $BC$, $AC$ and $AB$ at points $D$, $E$ and $F$, respectively. Let $E'$ be the reflection of point $E$ across line $DF$, and $F'$ be the reflection of point $F$ across line $DE$. Let line $EF$ intersect the circumcircle of triangle $AE'F'$ at points $X$ and $Y$. Prove that $DX=DY$. [i]Proposed by Márton Lovas, Budapest[/i]

Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$. [i]Proposed by Mohammad Javad Shabani[/i]

Let $\ell$ be tangent to the circle $k$ at $B$. Let $A$ be a point on $k$ and $P$ the foot of perpendicular from $A$ to $\ell$. Let $M$ be symmetric to $P$ with respect to $AB$. Find the set of all such points $M.$

Let $ABC$ be a triangle inscribed in circle $(O),$ with its altitudes $BE, CF$ intersect at orthocenter $H$ ($E \in AC, F \in AB$). Let $M$ be the midpoint of $BC, K$ be the orthogonal projection of $H$ on $AM$. $EF$ intersects $BC$ at $P$. Let $Q$ be the intersection of tangent of $(O)$ which passes through $A$ with $BC, T$ be the reflection of $Q$ through $P$. Prove that $\angle OKT = 90^o$.

A triangle $ABC$ has acute angles at $A$ and $B$. Isosceles triangles $ACD$ and $BCE$ with bases $AC$ and $BC$ are constructed externally to triangle $ABC$ such that $\angle ADC = \angle ABC$ and $\angle BEC = \angle BAC$. Let $S$ be the circumcenter of $\triangle ABC$. Prove that the length of the polygonal line $DSE$ equals the perimeter of triangle $ABC$ if and only if $\angle ACB$ is right.

Let $S=\{(x,y) : x \in \{0,1,2,3,4\}, y \in \{0,1,2,3,4,5\}$, and $(x,y) \neq (0,0) \}$. Let $T$ be the set of all right triangles whose vertices are in $S$. For every right triangle $t=\triangle ABC$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$, let $f(t)= \tan (\angle CBA)$. What is \[ \displaystyle \prod_{t \in T} f(t) \text{?} \] [asy] size((120)); dot((1,0)); dot((2,0)); dot((3,0)); dot((4,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((0,5)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((1,5)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((2,5)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((3,5)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); dot((4,5)); label("$\circ$", (0,0)); label("$S$", (-.7,2.5)); [/asy] $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ \frac{625}{144} \qquad \textbf{(C)}\ \frac{125}{24} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ \frac{625}{24}$

Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$. Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.

Let $I$ be the incenter of an acute-angled triangle $ABC$. Line $AI$ cuts the circumcircle of $BIC$ again at $E$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $J$ be the reflection of $I$ across $BC$. Show $D$, $J$ and $E$ are collinear.

Let $H$ be the orthocenter of $\triangle ABC$. The points $A_{1} \not= A$, $B_{1} \not= B$ and $C_{1} \not= C$ lie, respectively, on the circumcircles of $\triangle BCH$, $\triangle CAH$ and $\triangle ABH$ and satisfy $A_{1}H=B_{1}H=C_{1}H$. Denote by $H_{1}$, $H_{2}$ and $H_{3}$ the orthocenters of $\triangle A_{1}BC$, $\triangle B_{1}CA$ and $\triangle C_{1}AB$, respectively. Prove that $\triangle A_{1}B_{1}C_{1}$ and $\triangle H_{1}H_{2}H_{3}$ have the same orthocenter.

On sides $AB, AC, BC$ are points $M, X, Y$, respectively, such that $AX=MX$; $BY=MY$. $K$, $L$ are midpoints of $AY$ and $BX$. $O$ is circumcenter of $ABC$, $O_1$, $O_2$ are symmetric with $O$ with respect to $K$ and $L$. Prove that $X, Y, O_1, O_2$ are concyclic.

$\triangle{ABC}$ is isosceles$(AB=AC)$. Points $P$ and $Q$ exist inside the triangle such that $Q$ lies inside $\widehat{PAC}$ and $\widehat{PAQ} = \frac{\widehat{BAC}}{2}$. We also have $BP=PQ=CQ$.Let $X$ and $Y$ be the intersection points of $(AP,BQ)$ and $(AQ,CP)$ respectively. Prove that quadrilateral $PQYX$ is cyclic. [i](20 Points)[/i]

A point $X$ is marked on the base $BC$ of an isosceles $\triangle ABC$, and points $P$ and $Q$ are marked on the sides $AB$ and $AC$ so that $APXQ$ is a parallelogram. Prove that the point $Y$ symmetrical to $X$ with respect to line $PQ$ lies on the circumcircle of the $\triangle ABC$. [i]($5$ points)[/i]

Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\] (a) prove that $ ABC$ is a right-angled triangle, and (b) calculate $ \dfrac{BP}{CH}$. [i]Soewono, Bandung[/i]

Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

Let circles $ \Gamma $ and $ \omega $ are circumcircle and incircle of the triangle $ABC$, the incircle touches sides $BC,CA,AB$ at the points $A_1,B_1,C_1$. Let $A_2$ and $B_2$ lies the lines $A_1I$ and $B_1I$ ($A_1$ and $A_2$ lies different sides from $I$, $B_1$ and $B_2$ lies different sides from $I$) such that $IA_2=IB_2=R$. Prove that : (a) $AA_2=BB_2=IO$; (b) The lines $AA_2$ and $BB_2$ intersect on the circle $ \Gamma ;$

Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($ I,II$ and $ III$) equals three times the sum of the areas of the three inner squares ($ IV,V$ and $ VI$).