Given a quadrangle $ABCD$ and a point $M$ inside it such that $ABMD$ is a parallelogram. $ \angle CBM = \angle CDM$. Prove that the $ \angle ACD = \angle BCM$.

Let $D$ be an interior point of triangle $ABC$. Lines $BD$ and $CD$ intersect sides $AC$ and $AB$ at points $E$ and $F$, respectively. Points $X$ and $Y$ are on the plane such that $BFEX$ and $CEFY$ are parallelograms. Suppose lines $EY$ and $FX$ intersect at a point $T$ inside triangle $ABC$. Prove that points $B$, $C$, $E$, and $F$ are concyclic if and only if $\angle BAD = \angle CAT$.

A circle $K$ with radius $r$, a point $D$ on $K$, and a convex angle with vertex $S$ and rays $a$ and $b$ are given in the plane. Construct a parallelogram $ABCD$ such that $A$ and $B$ lie on $a$ and $b$ respectively, $SA+SB=r$, and $C$ lies on $K$.

In the parallelolgram A$BCD$ with the center $S$, let $O$ be the center of the circle of the inscribed triangle $ABD$ and let $T$ be the touch point with the diagonal $BD$. Prove that the lines $OS$ and $CT$ are parallel.

Given is a parallelogram $ABCD$, with $AB <AC <BC$. Points $E$ and $F$ are selected on the circumcircle $\omega$ of $ABC$ so that the tangenst to $\omega$ at these points pass through point $D$ and the segments $AD$ and $CE$ intersect. It turned out that $\angle ABF = \angle DCE$. Find the angle $\angle{ABC}$. A. Yakubov, S. Berlov

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.

The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.

Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let $E$ be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.

Given a parallelogram $ABCD$, join $A$ to the midpoints $E$ and $F$ of the opposite sides $BC$ and $CD$. $AE$ and $AF$ intersect the diagonal $BD$ in $M$ and $N$. Prove that $M$ and $N$ divide $BD$ into three equal parts.

A circle center $O$ is inscribed in the quadrilateral $ABCD$. $AB$ is parallel to and longer than $CD$ and has midpoint $M$. The line $OM$ meets $CD$ at $F$. $CD$ touches the circle at $E$. Show that $DE = CF$ iff $AB = 2CD$.

Two incongruent triangles $ABC$ and $XYZ$ are called a pair of [i]pals[/i] if they satisfy the following conditions: (a) the two triangles have the same area; (b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$. The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers. Determine if there are infinitely many pairs of triangles that are pals of each other.

Let $ABC$ be a triangle with circumcircle $k$. The points $A_1, B_1,$ and $C_1$ on $k$ are the midpoints of arcs $\widehat{BC}$ (not containing $A$), $\widehat{AC}$ (not containing $B$), and $\widehat{AB}$ (not containing $C$), respectively. The pairwise distinct points $A_2, B_2,$ and $C_2$ are chosen such that the quadrilaterals $AB_1A_2C_1, BA_1B_2C_1,$ and $CA_1C_2B_1$ are parallelograms. Prove that $k$ and the circumcircle of triangle $A_2B_2C_2$ have a common center. [b]Comment.[/b] Point $A_2$ can also be defined as the reflection of $A$ with respect to the midpoint of $B_1C_1$, and analogous definitions can be used for $B_2$ and $C_2$.

Let a quadrilateral $ABCD$ have an inscribed circle and let $K, L, M, N$ be the tangency points of the sides $AB, BC, CD$ and $DA$, respectively. Consider the orthocenters of each of the triangles $\vartriangle AKN, \vartriangle BLK, \vartriangle CML$ and $\vartriangle DNM$. Prove that these four points are the vertices of a parallelogram.

[b]points in plane[/b] set $A$ containing $n$ points in plane is given. a $copy$ of $A$ is a set of points that is made by using transformation, rotation, homogeneity or their combination on elements of $A$. we want to put $n$ $copies$ of $A$ in plane, such that every two copies have exactly one point in common and every three of them have no common elements. a) prove that if no $4$ points of $A$ make a parallelogram, you can do this only using transformation. ($A$ doesn't have a parallelogram with angle $0$ and a parallelogram that it's two non-adjacent vertices are one!) b) prove that you can always do this by using a combination of all these things. time allowed for this question was 1 hour and 30 minutes

An isosceles trapezoid $ABCD$ $(AB \parallel CD)$ is given. Points $E$ and $F$ lie on the sides $BC$ and $AD$, and the points $M$ and $N$ lie on the segment $EF$ such that $DF = BE$ and $FM = NE$. Let $K$ and $L$ be the foot of perpendicular lines from $M$ and $N$ to $AB$ and $CD$, respectively. Prove that $EKFL$ is a parallelogram. [i]Proposed by Mahdi Etesamifard[/i]

Let $A,B,C$ be fixed points in the plane , and $D$ be a variable point on the circle $ABC$, distinct from $A,B,C$ . Let $I_{A},I_{B},I_{C},I_{D}$ be the Simson lines of $A,B,C,D$ with respect to triangles $BCD,ACD,ABD,ABC$ respectively. Find the locus of the intersection points of the four lines $I_{A},I_{B},I_{C},I_{D}$ when point $D$ varies.

A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ respectively be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$ and $S_4$, $S_4$ and $S_1$. Prove that these four tangents determine a rhombus.

Quadrilateral $ABCD$ has $AB=BC=CD$, $\angle ABC=70^\circ$, and $\angle BCD=170^\circ$. What is the degree measure of $\angle BAD$? $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 80\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ 95$

Let $ ABC$ be an equilateral triangle of center $ O$, and $ M\in BC$. Let $ K,L$ be projections of $ M$ onto the sides $ AB$ and $ AC$ respectively. Prove that line $ OM$ passes through the midpoint of the segment $ KL$.

Four distinct lines $L_1,L_2,L_3,L_4$ are given in the plane: $L_1$ and $L_2$ are respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.

Suppose that $ABCD$ is a parallelogram such that $DAB>90$. Let the point $H$ to be on $AD$ such that $BH$ is perpendicular to $AD$. Let the point $M$ to be the midpoint of $AB$. Let the point $K$ to be the intersecting point of the line $DM$ with the circumcircle of $ADB$. Prove that $HKCD$ is concyclic.

Let $PQ$ be a line segment of constant length $\lambda$ taken on the side $BC$ of a triangle $ABC$ with the order $B,P,Q,C$, and let the lines through $P$ and $Q$ parallel to the lateral sides meet $AC$ at $P_1$ and $Q_1$ and $AB$ at $P_2$ and $Q_2$ respectively. Prove that the sum of the areas of the trapezoids $PQQ_1P_1$ and $PQQ_2P_2$ is independent of the position of $PQ$ on $BC.$

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 75\qquad\textbf{(E)}\ 90$

Given a triangle $ABC$. A point $X$ is chosen on a side $AC$. Some circle passes through $X$, touches the side $AC$ and intersects the circumcircle of triangle $ABC$ in points $M$ and $N$ such that the segment $MN$ bisects $BX$ and intersects sides $AB$ and $BC$ in points $P$ and $Q$. Prove that the circumcircle of triangle $PBQ$ passes through a fixed point different from $B$. [b]proposed by Sergej Berlov[/b]

Two circles $ O, O'$ having same radius meet at two points, $ A,B (A \not = B) $. Point $ P,Q $ are each on circle $ O $ and $ O' $ $(P \not = A,B ~ Q\not = A,B )$. Select the point $ R $ such that $ PAQR $ is a parallelogram. Assume that $ B, R, P, Q $ is cyclic. Now prove that $ PQ = OO' $.