Found problems: 1049
2017 Oral Moscow Geometry Olympiad, 3
Points $M$ and $N$ are the midpoints of sides $AB$ and $CD$, respectively of quadrilateral $ABCD$. It is known that $BC // AD$ and $AN = CM$. Is it true that $ABCD$ is parallelogram?
2024 AMC 10, 10
Quadrilateral $ABCD$ is a parallelogram, and $E$ is the midpoint of the side $\overline{AD}$. Let $F$ be the intersection of lines $EB$ and $AC$. What is the ratio of the area of quadrilateral $CDEF$ to the area of triangle $CFB$?
$\textbf{(A) } 5 : 4 \qquad \textbf{(B) } 4 : 3 \qquad \textbf{(C) } 3 : 2 \qquad \textbf{(D) } 5 : 3 \qquad \textbf{(E) } 2 : 1$
1983 Tournament Of Towns, (033) O2
(a) A regular $4k$-gon is cut into parallelograms. Prove that among these there are at least $k$ rectangles.
(b) Find the total area of the rectangles in (a) if the lengths of the sides of the $4k$-gon equal $a$.
(VV Proizvolov, Moscow)
2009 IMO Shortlist, 3
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]
2004 Olympic Revenge, 3
$ABC$ is a triangle and $\omega$ its incircle. Let $P,Q,R$ be the intersections with $\omega$ and the sides $BC,CA,AB$ respectively. $AP$ cuts $\omega$ in $P$ and $X$. $BX,CX$ cut $\omega$ in $M,N$ respectively. Show that $MR,NQ,AP$ are parallel or concurrent.
2006 India IMO Training Camp, 2
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
2011 National Olympiad First Round, 9
Let $ABCD$ be a convex quadrilateral with $m(\widehat{ADC}) = 90^{\circ}$. The line through $D$ which is parallel to $BC$ meets $AB$ at $E$. If $m(\widehat{DAC}) = m(\widehat{DAE})$, $|AB|=3$ and $|AC|=4$, then $|AE| = ?$
$\textbf{(A)}\ \frac56 \qquad\textbf{(B)}\ \frac13 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \frac34$
2007 IMO Shortlist, 4
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]
2010 Romania Team Selection Test, 4
Two circles in the plane, $\gamma_1$ and $\gamma_2$, meet at points $M$ and $N$. Let $A$ be a point on $\gamma_1$, and let $D$ be a point on $\gamma_2$. The lines $AM$ and $AN$ meet again $\gamma_2$ at points $B$ and $C$, respectively, and the lines $DM$ and $DN$ meet again $\gamma_1$ at points $E$ and $F$, respectively. Assume the order $M$, $N$, $F$, $A$, $E$ is circular around $\gamma_1$, and the segments $AB$ and $DE$ are congruent. Prove that the points $A$, $F$, $C$ and $D$ lie on a circle whose centre does not depend on the position of the points $A$ and $D$ on the respective circles, subject to the assumptions above.
[i]***[/i]
2010 India IMO Training Camp, 10
Let $ABC$ be a triangle. Let $\Omega$ be the brocard point. Prove that $\left(\frac{A\Omega}{BC}\right)^2+\left(\frac{B\Omega}{AC}\right)^2+\left(\frac{C\Omega}{AB}\right)^2\ge 1$
1984 Miklós Schweitzer, 8
[b]8.[/b] Among all point lattices on the plane intersecting every closed convex region of unit width, which on's fundamental parallelogram has the largest area? ([b]G.36[/b])
[L. Fejes-Tóth]
1988 AMC 12/AHSME, 6
A figure is an equiangular parallelogram if and only if it is a
$ \textbf{(A)}\ \text{rectangle}\qquad\textbf{(B)}\ \text{regular polygon}\qquad\textbf{(C)}\ \text{rhombus}\qquad\textbf{(D)}\ \text{square}\qquad\textbf{(E)}\ \text{trapezoid} $
2004 CHKMO, 3
Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.
2006 Estonia Math Open Junior Contests, 3
Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD
and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.
2005 Taiwan TST Round 3, 2
It is known that $\triangle ABC$ is an acute triangle. Let $C'$ be the foott of the perpendicular from $C$ to $AB$, and $D$, $E$ two distinct points on $CC'$. The feet of the perpendiculars from $D$ to $AC$ and $BC$ are $F$ and $G$, respectively. Show that if $DGEF$ is a parallelogram then $ABC$ is isosceles.
2001 Bulgaria National Olympiad, 2
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.
1989 IMO Shortlist, 32
The vertex $ A$ of the acute triangle $ ABC$ is equidistant from the circumcenter $ O$ and the orthocenter $ H.$ Determine all possible values for the measure of angle $ A.$
2007 Danube Mathematical Competition, 2
Let $ ABCD$ be an inscribed quadrilateral and let $ E$ be the midpoint of the diagonal $ BD$. Let $ \Gamma_1,\Gamma_2,\Gamma_3,\Gamma_4$ be the circumcircles of triangles $ AEB$, $ BEC$, $ CED$ and $ DEA$ respectively. Prove that if $ \Gamma_4$ is tangent to the line $ CD$, then $ \Gamma_1,\Gamma_2,\Gamma_3$ are tangent to the lines $ BC,AB,AD$ respectively.
1993 All-Russian Olympiad, 2
Segments $AB$ and $CD$ of length $1$ intersect at point $O$ and angle $AOC$ is equal to sixty degrees. Prove that $AC+BD \ge 1$.
2012 Saint Petersburg Mathematical Olympiad, 6
$ABCD$ is parallelogram. Line $l$ is perpendicular to $BC$ at $B$. Two circles passes through $D,C$, such that $l$ is tangent in points $P$ and $Q$. $M$ - midpoint $AB$.
Prove that $\angle DMP=\angle DMQ$
2011 District Olympiad, 1
On the sides $ AB,BC,CD,DA $ of the parallelogram $ ABCD, $ consider the points $ M,N,P, $ respectively, $ Q, $ such that $ \overrightarrow{MN} +\overrightarrow{QP} =\overrightarrow{AC} . $ Show that $ \overrightarrow{PN} +\overrightarrow{QM} = \overrightarrow{DB} . $
Estonia Open Junior - geometry, 1998.2.5
The points $E$ and $F$ divide the diagonal $BD$ of the convex quadrilateral $ABCD$ into three equal parts, i.e. $| BE | = | EF | = | F D |$. Line $AE$ interects side $BC$ at $X$ and line $AF$ intersects $DC$ at $Y$. Prove that:
a) if $ABCD$ is parallelogram then $X ,Y$ are the midpoints of $BC, DC$, respectively,
b) if the points $X , Y$ are the midpoints of $BC, DC$, respectively , then $ABCD$ is parallelogram
1996 Iran MO (3rd Round), 2
Let $ABCD$ be a parallelogram. Construct the equilateral triangle $DCE$ on the side $DC$ and outside of parallelogram. Let $P$ be an arbitrary point in plane of $ABCD$. Show that
\[PA+PB+AD \geq PE.\]
2020 Argentina National Olympiad Level 2, 3
Let $ABCD$ be a parallelogram with $\angle ABC = 105^\circ$. Inside the parallelogram, there is a point $E$ such that triangle $BEC$ is equilateral and $\angle CED = 135^\circ$. Let $K$ be the midpoint of side $AB$. Determine the measure of angle $\angle BKC$.
2005 All-Russian Olympiad, 3
We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$.