Given parallelogram $OABC$ in the coodinate with $O$ the origin and $A,B,C$ be lattice points. Prove that for all lattice point $P$ in the internal or boundary of $\triangle ABC$, there exists lattice points $Q,R$(can be the same) in the internal or boundary of $\triangle OAC$ with $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.

Let $ABCD$ and $A'B'C'D'$ be two arbitrary squares in the plane that are oriented in the same direction. Prove that the quadrilateral formed by the midpoints of $AA',BB',CC',DD'$ is a square.

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]

Let $ABCD$ be a parallelogram. Let $\omega_1$ be the circle passing through $D$ tangent to $AB$ at $A$. Let $\omega_2$ be the circle passing through $A$ tangent to $CD$ at $D$. The tangents from $B$ to $\omega_1$ touch it at $A$ and $P$. The tangents from $C$ to $\omega_2$ touch it at $D$ and $Q$. Lines $AP$ and $DQ$ intersect at $X$. The perpendicular bisector of $BC$ intersects $AD$ at $R$. Show that the circumcircles of triangles $\triangle PQX$, $\triangle BCR$ are concentric.

In the parallelogram $ABCD$, $\angle BAD =60 ^ \circ$. Let $E $ be the intersection point of the diagonals. The circle circumscribed to the triangle $ACD$ intersects the line $AB$ at the point $K$ (different from $A$), the line $BD$ at the point $P$ (different from $D$), and to the line $BC$ in $L$ (different from $C$). The line $EP$ intersects the circumscribed circle of the triangle $CEL$ at the points $E$ and $M$. Show that the triangles $KLM$ and $CAP$ are congruent.

Let $ABCD$ be a parallelogram and $M,N$ be points in the plane such that $C \in (AM)$ and $D \in (BN)$. Lines $NA,NC$ meet lines $MB,MD$ at points $E,F,G,H$. Show that points $E,F,G,H$ lie on a circle if and only if $ABCD$ is a rhombus.

Equilateral $\triangle ABC$ is inscribed in a circle of radius 2. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13$, and extend $\overline{AC}$ through $C$ to point $E$ so that $AE=11$. Through $D$, draw a line $l_1$ parallel to $\overline{AE}$, and through $E$, draw a line ${l}_2$ parallel to $\overline{AD}$. Let $F$ be the intersection of ${l}_1$ and ${l}_2$. Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A$. Given that the area of $\triangle CBG$ can be expressed in the form $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.

Let $ABCD$ be a rectangle with $BC=3AB$. Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$, then \[\angle DBC+\angle DPC = \angle DQC.\]

Let $ABCD$ be a quadrilateral that can be inscribed in a circle. Denote by $P$ the intersection point of lines $AD$ and $BC$, and by $Q$ the intersection point of lines $AB$ and $DC$. Let $E$ be the fourth vertex of the parallelogram $ABCE$, and $F$ the intersection of lines $CE$ is $PQ$. Prove that the points $D,E, F$, and $Q$ lie on the same circle.

On the sides of a parallelogram, squares are constructed outwards. Prove that the centers of these squares are vertices of a square.

Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$, denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions: [b]I.[/b] $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$. [b]II.[/b] Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$. (By $(AB)$ we denote vector $AB$)

In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.

Given points $ A $, $ B $, $ C $ and $ D $ that do not lie in the same plane. Draw a plane through the point $ A $ such that the orthogonal projection of the quadrilateral $ ABCD $ on this plane is a parallelogram.

$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

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 $A'$ and $B'$ be feet of altitudes from $A$ and $B$, respectively, in acute-angled triangle $ABC$ ($AC\not = BC$). Circle $k$ contains points $A'$ and $B'$ and touches segment $AB$ in $D$. If triangles $ADA'$ and $BDB'$ have the same area, prove that \[\angle A'DB'= \angle ACB.\]

Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals: [asy] size(7cm); pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5); pair G = intersectionpoints(B--F, D--C)[0]; pair E = intersectionpoints(A--C, B--F)[0]; draw(A--D--C--B--cycle); draw(A--C); draw(D--F--B); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$F$", F, N); label("$G$", G, NE); label("$E$", E, SE); //Credit to MSTang for the asymptote [/asy] $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 16$

$(BEL 2) (a)$ Find the equations of regular hyperbolas passing through the points $A(\alpha, 0), B(\beta, 0),$ and $C(0, \gamma).$ $(b)$ Prove that all such hyperbolas pass through the orthocenter $H$ of the triangle $ABC.$ $(c)$ Find the locus of the centers of these hyperbolas. $(d)$ Check whether this locus coincides with the nine-point circle of the triangle $ABC.$

$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$. Remark: the original question has missed the condition $ AB \neq AC$

In a triangle $ABC$ with $AB=AC$, $\omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $BA$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $OAD$ intersects the line $AC$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $DAEH$ is a parallelogram. Line $EH$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $GB$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $GJ$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle.

Let $ X,Y,Z$ be the midpoints of the small arcs $ BC,CA,AB$ respectively (arcs of the circumcircle of $ ABC$). $ M$ is an arbitrary point on $ BC$, and the parallels through $ M$ to the internal bisectors of $ \angle B,\angle C$ cut the external bisectors of $ \angle C,\angle B$ in $ N,P$ respectively. Show that $ XM,YN,ZP$ concur.

Complex number $z=\cos\theta+\text{i}\sin\theta(0\leq\theta\leq\pi)$. Points that three complex numbers $z,(1+\text{i})z,2\overline{z}$ refer to on complex plane are $P,Q,R$. When $P,Q,R$ are not collinear, $PQSR$ is a parallelogram. The longest distance between $S$ and the original point is________.

Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.

Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$, and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$, and $AHB$, respectively. Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $\triangle ABC$ is also the nine-point circle of $\triangle A_1B_1C_1$.