An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$. [i]Greece[/i]

Three circles $\mathcal{K}_1$, $\mathcal{K}_2$, $\mathcal{K}_3$ of radii $R_1,R_2,R_3$ respectively, pass through the point $O$ and intersect two by two in $A,B,C$. The point $O$ lies inside the triangle $ABC$. Let $A_1,B_1,C_1$ be the intersection points of the lines $AO,BO,CO$ with the sides $BC,CA,AB$ of the triangle $ABC$. Let $ \alpha = \frac {OA_1}{AA_1} $, $ \beta= \frac {OB_1}{BB_1} $ and $ \gamma = \frac {OC_1}{CC_1} $ and let $R$ be the circumradius of the triangle $ABC$. Prove that \[ \alpha R_1 + \beta R_2 + \gamma R_3 \geq R. \]

Let $ABCD$ be a parallelogram and $P$ a point in the plane. The line $BP$ intersects the circumcircle of $ABC$ again at $X$ and the line $DP$ intersects the circumcircle of $DAC$ again at $Y$. Let $M$ be the midpoint of the side $AC$. The point $N$ lies on the circumcircle of $PXY$ so that $MN$ is a tangent to this circle. Prove that the segments $MN$ and $AM$ have the same length. [i]Proposed by David Anghel[/i]

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$ [i]Proposed by Nazar Serdyuk, Ukraine[/i]

A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.

Does a parallelogram exist such that all pairwise meets of bisectors of its angles are situated outside it?

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]

Let $O, H$ be the circumcentor and the orthocenter of a scalene triangle $ABC$. Let $P$ be the reflection of $A$ w.r.t. $OH$, and $Q$ is a point on $\odot (ABC)$ such that $AQ, OH, BC$ are concurrent. Let $A'$ be a points such that $ABA'C$ is a parallelogram. Show that $A', H, P, Q$ are concylic. (ltf0501).

Let $O$ be the intersection point of the diagonals of a parallelogram $ABCD$ . Prove that if the line $BC$ is tangent to the circle passing through the points $A, B$, and $O$, then the line $CD$ is tangent to the circle passing through the points $B, C$ and $O$. (A Zaslavskiy)

Consider the parallelogram $ABCD$. A circle is drawn that passes through $A$ and intersects side $AD$ at $N$, side $AB$ at $M$ and diagonal $AC$ in $P$ such that points $A, M, N, P$ are different. Prove that $$AP\cdot AC = AM \cdot AB + AN \cdot AD.$$

Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{1}{3}DA$. What is the ratio of the area of triangle $DFE$ to the area of quadrilateral $ABEF$? $ \textbf{(A)}\ 1:2 \qquad\textbf{(B)}\ 1:3 \qquad\textbf{(C)}\ 1:5 \qquad\textbf{(D)}\ 1:6 \qquad\textbf{(E)}\ 1:7 $

Let $ ABCD$ be a trapezoid with $ AB$ and $ CD$ parallel, $ \angle D \equal{} 2 \angle B, AD \equal{} 5,$ and $ CD \equal{} 2.$ Then $ AB$ equals A. 7 B. 8 C. 13/2 D. 27/4 E. $ 5 \plus{} \frac{3 \sqrt{2}}{2}$

Prove that if all faces of a parallelepiped are equal parallelograms, they are rhombuses.

Let $ ABCDEF$ be a convex hexagon such that $ AD \equal{} BC \plus{} EF$, $ BE \equal{} AF \plus{} CD$, $ CF \equal{} DE \plus{} AB$. Prove that: \[ \frac {AB}{DE} \equal{} \frac {CD}{AF} \equal{} \frac {EF}{BC}. \]

Let $ABCD$ be a parallelogram with $\angle BAD<90^o$ and $AB> BC$ . The angle bisector of $BAD$ intersects line $CD$ at point $P$ and line $BC$ at point $Q$. Prove that the center of the circle circumscirbed around the triangle $CPQ$ is equidistant from points $B$ and $D$.

Let $ ABC$ be a triangle. The $ A$-excircle of triangle $ ABC$ has center $ O_{a}$ and touches the side $ BC$ at the point $ A_{a}$. The $ B$-excircle of triangle $ ABC$ touches its sidelines $ AB$ and $ BC$ at the points $ C_{b}$ and $ A_{b}$. The $ C$-excircle of triangle $ ABC$ touches its sidelines $ BC$ and $ CA$ at the points $ A_{c}$ and $ B_{c}$. The lines $ C_{b}A_{b}$ and $ A_{c}B_{c}$ intersect each other at some point $ X$. Prove that the quadrilateral $ AO_{a}A_{a}X$ is a parallelogram. [i]Remark.[/i] The $ A$[i]-excircle[/i] of a triangle $ ABC$ is defined as the circle which touches the segment $ BC$ and the extensions of the segments $ CA$ and $ AB$ beyound the points $ C$ and $ B$, respectively. The center of this circle is the point of intersection of the interior angle bisector of the angle $ CAB$ and the exterior angle bisectors of the angles $ ABC$ and $ BCA$. Similarly, the $ B$-excircle and the $ C$-excircle of triangle $ ABC$ are defined. [hide="Source of the problem"][i]Source of the problem:[/i] Theorem (88) in: John Sturgeon Mackay, [i]The Triangle and its Six Scribed Circles[/i], Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.[/hide]

Let $ABC$ be an acute triangle. Let $D$ and $E$ be the feet of the altitudes of the triangle $ABC$ from $A$ and $B$, respectively. Let $F$ be the reflection of the point $A$ over $BC$. Let $G$ be a point such that the quadrilateral $ABCG$ is a parallelogram. Show that the circumcircles of triangles $BCF$ , $ACG$ and $CDE$ are concurrent on a point different from $C$.

Consider two quadrilaterals $ A_1B_1C_1D_1,A_2B_2C_2D_2 $ and the points $ M,N,P,Q,E_1,F_1,E_2,F_2 $ representing the middle of the segments $ A_1A_2,B_1B_2,C_1C_2,D_1D_2,B_1D_1,A_1C_1,B_2D_2,A_2,C_2, $ respectively. Show that $ MNPQ $ is a parallelogram if and only if $ E_1F_1E_2F_2 $ is a parallelogram. [i]Cristinel Mortici[/i]

If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that \[4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)\]

Let $ABC$ be an acute triangle, $(O)$ be the circumcircle, and $AB<AC.$ Let $I$ be the midpoint of arc $BC$ (not containing $A$). $K$ lies on $AC,$ $K\ne C$ such that $IK=IC.$ $BK$ intersects $(O)$ at the second point $D,$ $D\ne B$ and intersects $AI$ at $E.$ $DI$ intersects $AC$ at $F.$ a) Prove that $EF=\frac{BC}{2}.$ b) $M$ lies on $DI$ such that $CM$ is parallel to $AD.$ $KM$ intersects $BC$ at $N.$ The circumcircle of triangle $BKN$ intersects $(O)$ at the second point $P.$ Prove that $PK$ passes through the midpoint of segment $AD.$

Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.

On a plane are given three non-collinear points $A, B, C$. We are given a disk of diameter different from that of the circle passing through $A, B, C$ large enough to cover all three points. Construct the fourth vertex of the parallelogram $ABCD$ using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).

The parallelogram $ABCD$ isn't a diamond. The ratio of the diagonal lengths $|AC|/|BD|$ equals to $k$. The $[AM)$ ray is symmetric to the $[AD)$ ray with respect to the $(AC)$ line. The $[BM)$ ray is symmetric to the $[BC)$ ray with respect to the $(BD)$ line. ($M$ point is those rays intersection.) Find the ratio $|AM|/|BM|$ .

Consider two triangles, $ABC$ and $DEF$, and any point $O$. We take any point $X$ in $\vartriangle ABC$ and any point $Y$ in $\vartriangle DEF$ and draw a parallelogram $OXY Z$. Prove that the locus of all possible points $Z$ form a polygon. How many sides can it have? Prove that its perimeter is equal to the sum of perimeters of the original triangles.