Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

Let $ AB$ be a diameter of a circle, $ C$ be any fixed point between $ A$ and $ B$ on this diameter, and $ Q$ be a variable point on the circumference of the circle. Let $ P$ be the point on the line determined by $ Q$ and $ C$ for which $ \frac{AC}{CB}\equal{}\frac{QC}{CP}$. Describe, with proof, the locus of the point $ P$.

Let $ABC$ be an acute triangle and let $A_{1}, B_{1}, C_{1}$ be the feet of its altitudes. The incircle of the triangle $A_{1}B_{1}C_{1}$ touches its sides at the points $A_{2}, B_{2}, C_{2}$. Prove that the Euler lines of triangles $ABC$ and $A_{2}B_{2}C_{2}$ coincide.

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

The circle $\zeta_{1}$ is inside the circle $\zeta_{2}$, and the circles touch each other at $A$. A line through $A$ intersects $\zeta_{1}$ also at $B$, and $\zeta_{2}$ also at $C$. The tangent to $\zeta_{1}$ at $B$ intersects $\zeta_{2}$ at $D$ and $E$. The tangents of $\zeta_{1}$ passing thorugh $C$ touch $\zeta_{2}$ at $F$ and $G$. Prove that $D$, $E$, $F$ and $G$ are concyclic.

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

Let $ABC$ to be a triangle and $\Gamma$ its circumcircle. Also, let $D, F, G$ and $E$, in this order, on the arc $BC$ which does not contain $A$ satisfying $\angle BAD = \angle CAE$ and $\angle BAF = \angle CAG$. Let $D`, F`, G`$ and $E`$ to be the intersections of $AD, AF, AG$ and $AE$ with $BC$, respectively. Moreover, $X$ is the intersection of $DF`$ with $EG`$, $Y$ is the intersection of $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ and $W$ is the intersection of $EF`$ with $DG`$. Prove that $X, Y$ and $A$ are collinear, such as $W, Z$ and $A$. Moreover, prove that $\angle BAX = \angle CAZ$.

Let $ ABCD$ be a convex quadrilateral such that $ AB$ and $ CD$ are not parallel and $ AB\equal{}CD$. The midpoints of the diagonals $ AC$ and $ BD$ are $ E$ and $ F$, respectively. The line $ EF$ meets segments $ AB$ and $ CD$ at $ G$ and $ H$, respectively. Show that $ \angle AGH \equal{} \angle DHG$.

Let $ABCD$ be a quadrilateral with $AB\parallel CD$ and $AC \perp BD$. Let $O$ be the intersection of $AC$ and $BD$. On the rays $(OA$ and $(OB$ we consider the points $M$ and $N$ respectively such that $\angle ANC = \angle BMD = 90^\circ$. We denote with $E$ the midpoint of the segment $MN$. Prove that a) $\triangle OMN \sim \triangle OBA$; b) $OE \perp AB$. [i]Claudiu-Stefan Popa[/i]

Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF \equal{} EG \equal{} EC$. Prove that $ \ell$ is the bisector of angle $ DAB$. [i]Author: Charles Leytem, Luxembourg[/i]

The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.

Consider a triangle $ABC$ with circumcenter $O$ and let $A_1,B_1,C_1$ be the midpoints of the sides $BC,AC,AB,$ respectively. Points $A_2,B_2,C_2$ are defined as $\overrightarrow{OA_2}=\lambda \cdot \overrightarrow{OA_1}, \ \overrightarrow{OB_2}=\lambda \cdot \overrightarrow{OB_1}, \ \overrightarrow{OC_2}=\lambda \cdot \overrightarrow{OC_1},$ where $\lambda >0.$ Prove that lines $AA_2,BB_2,CC_2$ are concurrent.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Suppose we have a square $ABCD$ and a point $S$ in the interior of this square. Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$. Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$. [asy] unitsize(3 cm); pair[] A, B, C, D; pair S; A[1] = (0,1); B[1] = (0,0); C[1] = (1,0); D[1] = (1,1); S = (0.3,0.6); A[0] = interp(S,A[1],2/3); B[0] = interp(S,B[1],2/3); C[0] = interp(S,C[1],2/3); D[0] = interp(S,D[1],2/3); draw(A[0]--B[0]--C[0]--D[0]--cycle); draw(A[1]--B[1]--C[1]--D[1]--cycle); draw(A[1]--S, dashed); draw(B[1]--S, dashed); draw(C[1]--S, dashed); draw(D[1]--S, dashed); dot("$A$", A[0], N); dot("$B$", B[0], SE); dot("$C$", C[0], SW); dot("$D$", D[0], SE); dot("$A'$", A[1], NW); dot("$B'$", B[1], SW); dot("$C'$", C[1], SE); dot("$D'$", D[1], NE); dot("$S$", S, dir(270)); [/asy]

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]

Let $P_0$, $P_1$, $P_2$ be three points on the circumference of a circle with radius $1$, where $P_1P_2 = t < 2$. For each $i \ge 3$, define $P_i$ to be the centre of the circumcircle of $\triangle P_{i-1} P_{i-2} P_{i-3}$. (1) Prove that the points $P_1, P_5, P_9, P_{13},\cdots$ are collinear. (2) Let $x$ be the distance from $P_1$ to $P_{1001}$, and let $y$ be the distance from $P_{1001}$ to $P_{2001}$. Determine all values of $t$ for which $\sqrt[500]{ \frac xy}$ is an integer.

Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.

Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively. Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively. Prove, that $I, O, H$ lies on one line.

Let $ ABCD$ be a convex quadrilateral and let $ P$ and $ Q$ be points in $ ABCD$ such that $ PQDA$ and $ QPBC$ are cyclic quadrilaterals. Suppose that there exists a point $ E$ on the line segment $ PQ$ such that $ \angle PAE \equal{} \angle QDE$ and $ \angle PBE \equal{} \angle QCE$. Show that the quadrilateral $ ABCD$ is cyclic. [i]Proposed by John Cuya, Peru[/i]

Let $\Gamma$ be the circumcircle of a triangle $ABC,$ and let $D$ and $E$ be two points different from the vertices on the sides $AB$ and $AC,$ respectively. Let $A'$ be the second point where $\Gamma$ intersects the bisector of the angle $BAC,$ and let $P$ and $Q$ be the second points where $\Gamma$ intersects the lines $A'D$ and $A'E,$ respectively. Let $R$ and $S$ be the second points of intersection of the lines $AA'$ and the circumcircles of the triangles $APD$ and $AQE,$ respectively. Show that the lines $DS, \: ER$ and the tangent line to $\Gamma$ through $A$ are concurrent.

Two circles $ \gamma_{1}$ and $ \gamma_{2}$ meet at points $ X$ and $ Y$. Consider the parallel through $ Y$ to the nearest common tangent of the circles. This parallel meets again $ \gamma_{1}$ and $ \gamma_{2}$ at $ A$, and $ B$ respectively. Let $ O$ be the center of the circle tangent to $ \gamma_{1},\gamma_{2}$ and the circle $ AXB$, situated outside $ \gamma_{1}$ and $ \gamma_{2}$ and inside the circle $ AXB.$ Prove that $ XO$ is the bisector line of the angle $ \angle{AXB}.$ [b]Radu Gologan[/b]