Olympiad problems

2007 Romania Team Selection Test, 2

Let $ A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that \[ [A_{1}A_{2}A_{3}] \equal{} [A_{2}A_{3}A_{4}] \equal{} [A_{3}A_{4}A_{5}] \equal{} [A_{4}A_{5}A_{1}] \equal{} [A_{5}A_{1}A_{2}].\] Prove that there exists a point $ M$ in the plane of the pentagon such that \[ [A_{1}MA_{2}] \equal{} [A_{2}MA_{3}] \equal{} [A_{3}MA_{4}] \equal{} [A_{4}MA_{5}] \equal{} [A_{5}MA_{1}].\] Here $ [XYZ]$ stands for the area of the triangle $ \Delta XYZ$.

2014 Puerto Rico Team Selection Test, 1

Tags: geometry , parallelogram , perpendicular bisector

Let $ABCD$ be a parallelogram with $AB>BC$ and $\angle DAB$ less than $\angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at the point $M$ lying on the extension of $AD$. If $\angle MCD=15^{\circ}$, find the measure of $\angle ABC$

2010 Cuba MO, 8

Tags: geometry , parallelogram , locus

Let $ABCDE$ be a convex pentagon that has $AB < BC$, $AE <ED$ and $AB + CD + EA = BC + DE$. Variable points $F,G$ and $H$ are taken that move on the segments $BC$, $CD$ and $OF$ respectively . $B'$ is defined as the projection of $B$ on $AF$, $C'$ as the projection of $C$ on $FG$, $D'$ as the projection of $D$ on $GH$ and $E'$ as the projection of $E$ onto $HA$. Prove that there is at least one quadrilateral $B'C'D'E'$ when $F,G$ and $H$ move on their sides, which is a parallelogram.

1996 North Macedonia National Olympiad, 1

Tags: geometry , parallelogram , equal angles , perpendicular

Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$. Prove that $\angle DEA = \angle CEB$.

1966 IMO Shortlist, 37

Tags: geometry , parallelogram , cyclic quadrilateral

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.

2006 China Team Selection Test, 1

Tags: parallelogram , geometry

Let the intersections of $\odot O_1$ and $\odot O_2$ be $A$ and $B$. Point $R$ is on arc $AB$ of $\odot O_1$ and $T$ is on arc $AB$ on $\odot O_2$. $AR$ and $BR$ meet $\odot O_2$ at $C$ and $D$; $AT$ and $BT$ meet $\odot O_1$ at $Q$ and $P$. If $PR$ and $TD$ meet at $E$ and $QR$ and $TC$ meet at $F$, then prove: $AE \cdot BT \cdot BR = BF \cdot AT \cdot AR$.

2001 All-Russian Olympiad, 3

Tags: parallelogram , circumcircle , geometry

A point $K$ is taken inside parallelogram $ABCD$ so that the midpoint of $AD$ is equidistant from $K$ and $C$, and the midpoint of $CD$ is equidistant form $K$ and $A$. Let $N$ be the midpoint of $BK$. Prove that the angles $NAK$ and $NCK$ are equal.

1972 IMO Longlists, 43

Tags: parallelogram , rectangle , trapezoid , pythagorean theorem , perpendicular bisector , geometry

A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.

2004 Germany Team Selection Test, 1

Tags: parallelogram , trigonometry , angle bisector , geometry

The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

Champions Tournament Seniors - geometry, 2018.3

Tags: geometry , parallelogram , equal angles , equal segments

The vertex $F$ of the parallelogram $ACEF$ lies on the side $BC$ of parallelogram $ABCD$. It is known that $AC = AD$ and $AE = 2CD$. Prove that $\angle CDE = \angle BEF$.

2001 All-Russian Olympiad Regional Round, 10.2

Tags: geometry , parallel , parallelogram

In parallelogram $ABCD$, point $K$ is marked on diagonal $AC$. Circle $s_1$ passes through point $K$ and touches lines $AB$ and $AD$ ($s_1$ intersects the diagonal $AC$ for the second time on the segment $AK$). Circle $s_2$ passes through point $K$ and touches lines $CB$ and $CD$ ($s_2$ intersects for the second time diagonal $AC$ on segment $KC$). Prove that for all positions of the point $K$ on the diagonal $AC$, the straight lines connecting the centers of circles $ s_1$ and $s_2$, will be parallel to each other.

2011 China Girls Math Olympiad, 8

Tags: geometry , geometric transformation , homothety , incenter , parallelogram , projective geometry , trigonometry

The $A$-excircle $(O)$ of $\triangle ABC$ touches $BC$ at $M$. The points $D,E$ lie on the sides $AB,AC$ respectively such that $DE\parallel BC$. The incircle $(O_1)$ of $\triangle ADE$ touches $DE$ at $N$. If $BO_1\cap DO=F$ and $CO_1\cap EO=G$, prove that the midpoint of $FG$ lies on $MN$.

2000 Dutch Mathematical Olympiad, 3

Tags: geometry , parallelogram , trapezoid , similar triangles

Isosceles, similar triangles $QPA$ and $SPB$ are constructed (outwards) on the sides of parallelogram $PQRS$ (where $PQ = AQ$ and $PS = BS$). Prove that triangles $RAB$, $QPA$ and $SPB$ are similar.

1989 Romania Team Selection Test, 3

Tags: geometry , parallelogram , min , geometric inequalities , inequalities

(a) Find the point $M$ in the plane of triangle $ABC$ for which the sum $MA + MB+ MC$ is minimal. (b) Given a parallelogram $ABCD$ whose angles do not exceed $120^o$, determine $min \{MA+ MB+NC+ND+ MN | M,N$ are in the plane $ABCD\}$ in terms of the sides and angles of the parallelogram.

2013 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , parallelogram , incircle , parallel

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.

2009 Junior Balkan Team Selection Test, 3

Tags: parallelogram , perpendicular bisector , geometry

Let $ ABCD$ be a convex quadrilateral, such that $ \angle CBD\equal{}2\cdot\angle ADB, \angle ABD\equal{}2\cdot\angle CDB$ and $ AB\equal{}CB$. Prove that quadrilateral $ ABCD$ is a kite.

1998 All-Russian Olympiad Regional Round, 8.2

Tags: parallelogram , trigonometry , geometry

Given a parallelogram ABCD, let M and N be the midpoints of the sides BC and CD. Can the lines AM, AN divide the angle BAD into three equal angles?

2007 China Team Selection Test, 1

Tags: circumcircle , parallelogram , perpendicular bisector , geometry

Let convex quadrilateral $ ABCD$ be inscribed in a circle centers at $ O.$ The opposite sides $ BA,CD$ meet at $ H$, the diagonals $ AC,BD$ meet at $ G.$ Let $ O_{1},O_{2}$ be the circumcenters of triangles $ AGD,BGC.$ $ O_{1}O_{2}$ intersects $ OG$ at $ N.$ The line $ HG$ cuts the circumcircles of triangles $ AGD,BGC$ at $ P,Q$, respectively. Denote by $ M$ the midpoint of $ PQ.$ Prove that $ NO \equal{} NM.$

1998 Turkey MO (2nd round), 2

Tags: geometric transformation , reflection , parallelogram , angle bisector , hjelmslev , geometry

Variable points $M$ and $N$ are considered on the arms $\left[ OX \right.$ and $\left[ OY \right.$ , respectively, of an angle $XOY$ so that $\left| OM \right|+\left| ON \right|$ is constant. Determine the locus of the midpoint of $\left[ MN \right]$.

2015 Saint Petersburg Mathematical Olympiad, 4

Tags: geometry , circumcircle , parallelogram

$ABCD$ is convex quadrilateral. Circumcircle of $ABC$ intersect $AD$ and $DC$ at points $P$ and $Q$. Circumcircle of $ADC$ intersect $AB$ and $BC$ at points $S$ and $R$. Prove that if $PQRS$ is parallelogram then $ABCD$ is parallelogram

2014 Abels Math Contest (Norwegian MO) Final, 2

Tags: geometry , angle bisector , concurrency , parallelogram

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 $\angle BAD$.

2007 Switzerland - Final Round, 6

Tags: geometry , parallelogram , equal circles

Three equal circles $k_1, k_2, k_3$ intersect non-tangentially at a point $P$. Let $A$ and $B$ be the centers of circles $k_1$ and $k_2$. Let $D$ and $C$ be the intersection of $k_3$ with $k_1$ and $k_2$ respectively, which is different from $P$. Show that $ABCD$ is a parallelogram.