Found problems: 1049
1992 IMO Longlists, 65
If $A, B, C$, and $D$ are four distinct points in space, prove that there is a plane $P$ on which the orthogonal projections of $A, B, C$, and $D$ form a parallelogram (possibly degenerate).
2013 Polish MO Finals, 3
Given is a quadrilateral $ABCD$ in which we can inscribe circle. The segments $AB, BC, CD$ and $DA$ are the diameters of the circles $o1, o2, o3$ and $o4$, respectively. Prove that there exists a circle tangent to all of the circles $o1, o2, o3$ and $o4$.
Durer Math Competition CD Finals - geometry, 2012.C3
Given a convex quadrilateral whose opposite sides are not parallel, and giving an internal point $P$. Find a parallelogram whose vertices are on the side lines of the rectangle and whose center is $P$. Give a method by which we can construct it (provided there is one).
[img]https://1.bp.blogspot.com/-t4aCJza0LxI/X9j1qbSQE4I/AAAAAAAAMz4/V9pr7Cd22G4F320nyRLZMRnz18hMw9NHQCLcBGAsYHQ/s0/2012%2BDurer%2BC3.png[/img]
2001 Baltic Way, 6
The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $AB\parallel EC$ and $AC\parallel ED$. The line tangent to the circle $c$ at $E$ meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$. Prove that $|AC|=|PQ|$.
2012 Serbia National Math Olympiad, 1
Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that \[\angle AED=\angle PEB.\]
2010 Germany Team Selection Test, 1
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]
1999 South africa National Olympiad, 3
The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$.
2011 AMC 8, 25
A circle with radius $1$ is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
[asy]
filldraw((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray,black);
filldraw(Circle((0,0),1), mediumgray,black);
filldraw((-1,0)--(0,1)--(1,0)--(0,-1)--cycle,white,black);[/asy]
$ \textbf{(A)}\ \frac{1}2\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ \frac{3}2\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{5}2 $
1991 Balkan MO, 1
Let $ABC$ be an acute triangle inscribed in a circle centered at $O$. Let $M$ be a point on the small arc $AB$ of the triangle's circumcircle. The perpendicular dropped from $M$ on the ray $OA$ intersects the sides $AB$ and $AC$ at the points $K$ and $L$, respectively. Similarly, the perpendicular dropped from $M$ on the ray $OB$ intersects the sides $AB$ and $BC$ at $N$ and $P$, respectively. Assume that $KL=MN$. Find the size of the angle $\angle{MLP}$ in terms of the angles of the triangle $ABC$.
2011 Germany Team Selection Test, 2
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]
2008 Sharygin Geometry Olympiad, 19
(V.Protasov, 10-11) Given parallelogram $ ABCD$ such that $ AB \equal{} a$, $ AD \equal{} b$. The first circle has its center at vertex $ A$ and passes through $ D$, and the second circle has its center at $ C$ and passes through $ D$. A circle with center $ B$ meets the first circle at points $ M_1$, $ N_1$, and the second circle at points $ M_2$, $ N_2$. Determine the ratio $ M_1N_1/M_2N_2$.
2015 District Olympiad, 1
Consider the parallelogram $ ABCD, $ whose diagonals intersect at $ O. $ The bisector of the angle $ \angle DAC $ and that of $ \angle DBC $ intersect each other at $ T. $ Moreover, $ \overrightarrow{TD} +\overrightarrow{TC} =\overrightarrow{TO} . $
Find the angles of the triangle $ ABT. $
2005 Iran Team Selection Test, 2
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that:
\[PX || AC \ , \ PY ||AB \]
Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$
2003 Indonesia MO, 2
Let $ABCD$ be a quadrilateral, and $P,Q,R,S$ are the midpoints of $AB, BC, CD, DA$ respectively. Let $O$ be the intersection between $PR$ and $QS$. Prove that $PO = OR$ and $QO = OS$.
1992 Hungary-Israel Binational, 4
We are given a convex pentagon $ABCDE$ in the coordinate plane such that $A$, $B$, $C$, $D$, $E$ are lattice points. Let $Q$ denote the convex pentagon bounded by the five diagonals of the pentagon $ABCDE$ (so that the vertices of $Q$ are the interior points of intersection of diagonals of the pentagon $ABCDE$). Prove that there exists a lattice point inside of $Q$ or on the boundary of $Q$.
1983 IMO Longlists, 48
Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.
2004 239 Open Mathematical Olympiad, 7
Given an isosceles triangle $ABC$ (with $AB=BC$). 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 the circumcentre of triangle $ABC$.
[b]proposed by Sergej Berlov[/b]
2011 Peru MO (ONEM), 3
Let $ABC$ be a right triangle, right in $B$. Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$. Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$, respectively. Prove that $UV$ is perpendicular to $AC$.
1998 Hungary-Israel Binational, 2
On the sides of a convex hexagon $ ABCDEF$ , equilateral triangles are constructd in its exterior. Prove that the third vertices of these six triangles are vertices of a regular hexagon if and only if the initial hexagon is [i]affine regular[/i]. (A hexagon is called affine regular if it is centrally symmetric and any two opposite sides are parallel to the diagonal determine by the remaining two vertices.)
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]
2010 Danube Mathematical Olympiad, 2
Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.
1998 Tournament Of Towns, 2
$ABCD$ is a parallelogram. A point $M$ is found on the side $AB$ or its extension such that $\angle MAD = \angle AMO$ where $O$ is the intersection point of the diagonals of the parallelogram. Prove that $MD = MG$.
(M Smurov)
2014 Israel National Olympiad, 3
Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.
1986 India National Olympiad, 6
Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
Estonia Open Junior - geometry, 2019.1.5
Point $M$ lies on the diagonal $BD$ of parallelogram $ABCD$ such that $MD = 3BM$. Lines $AM$ and $BC$ intersect in point $N$. What is the ratio of the area of triangle $MND$ to the area of parallelogram $ABCD$?