We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$. [i]Proposed by Ali Zamani[/i]

In triangle $ ABC$ the bisector of angle $ BCA$ intersects the circumcircle again at $ R$, the perpendicular bisector of $ BC$ at $ P$, and the perpendicular bisector of $ AC$ at $ Q$. The midpoint of $ BC$ is $ K$ and the midpoint of $ AC$ is $ L$. Prove that the triangles $ RPK$ and $ RQL$ have the same area. [i]Author: Marek Pechal, Czech Republic[/i]

Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.

Consider acute $\triangle ABC$ with altitudes $AA_1, BB_1$ and $CC_1$ ($A_1 \in BC,B_1 \in AC,C_1 \in AB$). A point $C' $ on the extension of $B_1A_1$ beyond $A_1$ is such that $A_1C' = B_1C_1$. Analogously, a point $B'$ on the extension of A$_1C_1$ beyond $C_1$ is such that $C_1B' = A_1B_1$ and a point $A' $ on the extension of $C_1B_1$ beyond $B_1$ is such that $B_1A' = C_1A_1$. Denote by $A'', B'', C''$ the symmetric points of $A' , B' , C'$ with respect to $BC, CA$ and $AB$ respectively. Prove that if $R, R'$ and R'' are circumradiii of $\triangle ABC, \triangle A'B'C'$ and $\triangle A''B''C''$, then $R, R'$ and $R'' $ are sidelengths of a triangle with area equals one half of the area of $\triangle ABC$.

The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$ so that $CD = BC$. The side $CA$ is extended beyond $A$ to $E$ so that $AE = 2CA$. Prove that, if $AD=BE$, then the triangle $ABC$ is right-angled.

Let $M$ be a point on the altitude $CD$ of an acute-angled triangle $ABC$, and $K$ and $L$ the orthogonal projections of $M$ on $AC$ and $BC$. Suppose that the incenter and circumcenter of the triangle lie on the segment $KL$. (a) Prove that $CD=R+r$, where $R$ and $r$ are the circumradius and inradius, respectively. (b) Find the minimum value of the ratio $CM:CD$.

Suppose that $\triangle ABC$ is an equilateral triangle of side length $s$, with the property that there is a unique point $P$ inside the triangle such that $AP = 1$, $BP = \sqrt{3}$, and $CP = 2$. What is $s?$ $\textbf{(A) } 1 + \sqrt{2} \qquad \textbf{(B) } \sqrt{7} \qquad \textbf{(C) } \frac{8}{3} \qquad \textbf{(D) } \sqrt{5 + \sqrt{5}} \qquad \textbf{(E) } 2\sqrt{2}$

In $\triangle ABC$, points $P$ and $Q$ lie on sides $AB$ and $AC$, respectively, such that the circumcircle of $\triangle APQ$ is tangent to $BC$ at $D$. Let $E$ lie on side $BC$ such that $BD = EC$. Line $DP$ intersects the circumcircle of $\triangle CDQ$ again at $X$, and line $DQ$ intersects the circumcircle of $\triangle BDP$ again at $Y$. Prove that $D$, $E$, $X$, and $Y$ are concyclic.

Find all point $M$ lying into given acute-angled triangle $ABC$ and such that the area of the triangle with vertices on the feet of the perpendiculars drawn from $M$ to the lines $BC$, $CA$, $AB$ is maximal. [i]H. Lesov[/i]

In triangle $ABC,\angle BAC = 60^o, AB = 3a$ and $AC = 4a, (a > 0)$. Let $M$ be point on the segment $AB$ such that $AM =\frac13 AB, N$ be point on the side $AC$ such that $AN =\frac12AC$. Let $I$ be midpoint of $MN$. Determine the length of $BI$. A. $\frac{a\sqrt2}{19}$ B. $\frac{2a}{\sqrt{19}}$ C. $\frac{19a\sqrt{19}}{2}$ D. $\frac{19a}{\sqrt2}$ E. $\frac{a\sqrt{19}}{2}$

Given triangle $ ABC $ with points $ M $ and $ N $ are in the sides $ AB $ and $ AC $ respectively. If $ \dfrac{BM}{MA} +\dfrac{CN}{NA} = 1 $ , then prove that the centroid of $ ABC $ lies on $ MN $ .

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Let $\triangle ABC$ be an acute-angled triangle, and let $D$ be the foot of the altitude from $C.$ The angle bisector of $\angle ABC$ intersects $CD$ at $E$ and meets the circumcircle $\omega$ of triangle $\triangle ADE$ again at $F.$ If $\angle ADF = 45^{\circ}$, show that $CF$ is tangent to $\omega .$