Found problems: 1049
2024 Bangladesh Mathematical Olympiad, P5
Consider $\triangle XPQ$ and $\triangle YPQ$ such that $X$ and $Y$ are on the opposite sides of $PQ$ and the circumradius of $\triangle XPQ$ and the circumradius of $\triangle YPQ$ are the same. $I$ and $J$ are the incenters of $\triangle XPQ$ and $\triangle YPQ$ respectively. Let $M$ be the midpoint of $PQ$. Suppose $I, M, J$ are collinear. Prove that $XPYQ$ is a parallelogram.
2009 Sharygin Geometry Olympiad, 21
The opposite sidelines of quadrilateral $ ABCD$ intersect at points $ P$ and $ Q$. Two lines passing through these points meet the side of $ ABCD$ in four points which are the vertices of a parallelogram. Prove that the center of this parallelogram lies on the line passing through the midpoints of diagonals of $ ABCD$.
1966 IMO Longlists, 37
Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent.
[b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.
2010 Contests, 3
In plane,let a circle $(O)$ and two fixed points $B,C$ lies in $(O)$
such that $BC$ not is the diameter.Consider a point $A$ varies in
$(O)$ such that $A\neq B,C$ and $AB\neq AC$.Call $D$ and $E$
respective is intersect of $BC$ and internal and external bisector
of $\widehat{BAC}$,$I$ is midpoint of $DE$.The line that pass through
orthocenter of $\triangle ABC$
and perpendicular with $AI$ intersects $AD,AE$ respective at $M,N$.
1/Prove that $MN$ pass through a fixed point
2/Determint the place of $A$ such that $S_{AMN}$ has maxium value
2011 Portugal MO, 2
The point $P$, inside the triangle $[ABC]$, lies on the perpendicular bisector of $[AB]$. $Q$ and $R$ points, exterior to the triangle, they are such that $ [BPA], [BQC]$ and $[CRA]$ are similar triangles. Shows that $[PQCR]$ is a parallelogram.
[img]https://cdn.artofproblemsolving.com/attachments/f/5/6e036b127f8a013794b8246cbb1544e7280d4a.png[/img]
Ukrainian From Tasks to Tasks - geometry, 2010.9
On the sides $AB, BC, CD$ and $DA$ of the parallelogram $ABCD$ marked the points $M, N, K$ and $F$. respectively. Is it possible to determine, using only compass, whether the area of ​​the quadrilateral $MNKF$ is equal to half the area of ​​the parallelogram $ABCD$?
2008 USA Team Selection Test, 7
Let $ ABC$ be a triangle with $ G$ as its centroid. Let $ P$ be a variable point on segment $ BC$. Points $ Q$ and $ R$ lie on sides $ AC$ and $ AB$ respectively, such that $ PQ \parallel AB$ and $ PR \parallel AC$. Prove that, as $ P$ varies along segment $ BC$, the circumcircle of triangle $ AQR$ passes through a fixed point $ X$ such that $ \angle BAG = \angle CAX$.
2005 Romania National Olympiad, 1
Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$, and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$. Let $F= AM \cap BC$. Prove that:
a) $DE=AF$;
b) $AD\cdot AB = DE\cdot DM$.
[i]Daniela and Marius Lobaza, Timisoara[/i]
2000 National Olympiad First Round, 5
$[BD]$ is a median of $\triangle ABC$. $m(\widehat{ABD})=90^\circ$, $|AB|=2$, and $|AC|=6$. $|BC|=?$
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 3\sqrt2
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4\sqrt2
\qquad\textbf{(E)}\ 2\sqrt6
$
2014 Online Math Open Problems, 4
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he has five total emons, let $P$ be the product of the $\binom 52 = 10$ distances between the $10$ pairs of emons. Find the greatest possible value of $P^2$.
[i]Proposed by Yang Liu[/i]
2022 Rioplatense Mathematical Olympiad, 4
Let $ABCD$ be a parallelogram and $M$ is the intersection of $AC$ and $BD$. The point $N$ is inside of the $\triangle AMB$ such that $\angle AND=\angle BNC$. Prove that $\angle MNC=\angle NDA$ and $\angle MND=\angle NCB$.
2011 All-Russian Olympiad, 2
On side $BC$ of parallelogram $ABCD$ ($A$ is acute) lies point $T$ so that triangle $ATD$ is an acute triangle. Let $O_1$, $O_2$, and $O_3$ be the circumcenters of triangles $ABT$, $DAT$, and $CDT$ respectively. Prove that the orthocenter of triangle $O_1O_2O_3$ lies on line $AD$.
2013 Tournament of Towns, 2
On the sides of triangle $ABC$, three similar triangles are constructed with triangle $YBA$ and triangle $ZAC$ in the exterior and triangle $XBC$ in the interior. (Above, the vertices of the triangles are ordered so that the similarities take vertices to corresponding vertices, for example, the similarity between triangle $YBA$ and triangle $ZAC$ takes $Y$ to $Z, B$ to $A$ and $A$ to $C$). Prove that $AYXZ$ is a parallelogram
2023 All-Russian Olympiad Regional Round, 10.8
The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$
2019 Iran Team Selection Test, 3
Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$.
[i]Proposed by Siamak Ahmadpour[/i]
2019 Belarusian National Olympiad, 10.8
Call a polygon on a Cartesian plane to be[i]integer[/i] if all its vertices are integer. A convex integer $14$-gon is cut into integer parallelograms with areas not greater than $C$.
Find the minimal possible $C$.
[i](A. Yuran)[/i]
2008 Sharygin Geometry Olympiad, 4
(F.Nilov, A.Zaslavsky) Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA \equal{} \angle C_0CB$.
2001 JBMO ShortLists, 12
Consider the triangle $ABC$ with $\angle A= 90^{\circ}$ and $\angle B \not= \angle C$. A circle $\mathcal{C}(O,R)$ passes through $B$ and $C$ and intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. Let $S$ be the foot of the perpendicular from $A$ to $BC$ and let $K$ be the intersection point of $AS$ with the segment $DE$. If $M$ is the midpoint of $BC$, prove that $AKOM$ is a parallelogram.
2009 Grand Duchy of Lithuania, 4
A triangle $ ABC$ has an obtuse angle at $ B$. The perpindicular at $ B$ to $ AB$ meets $ AC$ at $ D$, and $ |CD| \equal{} |AB|$.
Prove that $ |AD|^2 \equal{} |AB|.|BC|$ if and only if $ \angle CBD \equal{} 30^\circ$.
2006 Grigore Moisil Urziceni, 1
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]
2005 Baltic Way, 12
Let $ABCD$ be a convex quadrilateral such that $BC=AD$. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. The lines $AD$ and $BC$ meet the line $MN$ at $P$ and $Q$, respectively. Prove that $CQ=DP$.
2016 Regional Olympiad of Mexico Center Zone, 4
Let $A$ be one of the two points where the circles whose centers are the points $M$ and $N$ intersect. The tangents in $A$ to such circles intersect them again in $B$ and $C$, respectively. Let $P$ be a point such that the quadrilateral $AMPN$ is a parallelogram. Show that $P$ is the circumcenter of triangle $ABC$.
1969 IMO Shortlist, 2
$(BEL 2) (a)$ 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 $ABC.$
2019 Saudi Arabia JBMO TST, 1
Let $E$ be a point lies inside the parallelogram $ABCD$ such that $\angle BCE = \angle BAE$.
Prove that the circumcenters of triangles $ABE,BCE,CDE,DAE$ are concyclic.
2013 Bundeswettbewerb Mathematik, 2
A parallelogram of paper with sides $25$ and $10$ is given. The distance between the longer sides is $6$. The paper should be cut into exactly two parts in such a way that one can stick both the pieces together and fold it in a suitable manner to form a cube of suitable edge length without any further cuts and overlaps. Show that it is really possible and describe such a fragmentation.