Olympiad problems

2011 Sharygin Geometry Olympiad, 5

A line passing through vertex $A$ of regular triangle $ABC$ doesn’t intersect segment $BC$. Points $M$ and $N$ lie on this line, and $AM = AN = AB$ (point $B$ lies inside angle $MAC$). Prove that the quadrilateral formed by lines $AB, AC, BN, CM$ is cyclic.

2004 Pan African, 3

Tags: geometry , circumcircle , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.

2003 Hong kong National Olympiad, 3

Tags: parallelogram , analytic geometry , cyclic quadrilateral , geometry

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.

2018 Iran MO (1st Round), 20

Tags: geometry , cyclic quadrilateral

In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$ and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\angle PDF$ in degrees?

2021 Cono Sur Olympiad, 2

Tags: cyclic quadrilateral , geometry , triangle -incenter , circumcircle

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

2003 Bundeswettbewerb Mathematik, 3

Tags: cyclic quadrilateral , perpendicular bisector , geometry

Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$. Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively. Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.

Ukraine Correspondence MO - geometry, 2012.10

Tags: geometry , equal segments , cyclic quadrilateral

The diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$ intersect at a point O. It is known that $\angle BAD = 60^o$ and $AO = 3OC$. Prove that the sum of some two sides of a quadrilateral is equal to the sum of the other two sides.

2016 China Team Selection Test, 3

Tags: geometry , cyclic quadrilateral , incenter

In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.

1979 Czech And Slovak Olympiad IIIA, 3

Tags: geometry , geometric inequality , square , cyclic quadrilateral

If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]

2016 Puerto Rico Team Selection Test, 3

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral. Let $ P$ be the intersection of the lines $BC$ and $AD$. Line $AC$ cuts the circumscribed circle of the triangle $BDP$ in $S$ and $T$, with $S$ between $ A$ and $C$. The line $BD$ intersects the circumscribed circle of the triangle $ACP$ in $U$ and $V$, with $U$ between $ B$ and $D$. Prove that $PS = PT = PU = PV$.

2017 NIMO Problems, 5

Tags: geometry , cyclic quadrilateral

Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Points $D$ and $E$ are chosen on $AC$ and $AB$, respectively, such that quadrilateral $BCDE$ is cyclic and when the triangle is folded along segment $DE$, point $A$ lies on side $BC$. If the length of $DE$ can be expressed as $\tfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $100m+n$. [i]Proposd by Joseph Heerens[/i]

Ukrainian TYM Qualifying - geometry, 2011.14

Tags: geometry , concurrency , concyclic , symmetry , cyclic quadrilateral

Given a quadrilateral $ABCD$, inscribed in a circle $\omega$ such that $AB=AD$ and $CB=CD$ . Take the point $P \in \omega$. Let the vertices of the quadrilateral $Q_1Q_2Q_3Q_4$ be symmetric to the point P wrt the lines $AB$, $BC$, $CD$, and $DA$, respectively. a) Prove that the points symmetric to the point $P$ wrt lines $Q_1Q_22, Q_2Q_3, Q_3Q_4$ and $Q_4Q_1$, lie on one line. b) Prove that when the point $P$ moves in a circle $\omega$, then all such lines pass through one common point.

2024 Bangladesh Mathematical Olympiad, P2

Tags: geometry , cyclic quadrilateral , power of a point

In a cyclic quadrilateral $ABCD$, the diagonals intersect at $E$. $F$ and $G$ are on chord $AC$ and chord $BD$ respectively such that $AF = BE$ and $DG = CE$. Prove that, $A, G, F, D$ lie on the same circle.

2010 Peru IMO TST, 8

Tags: geometry , cyclic quadrilateral , projective geometry , tangent , power of a point

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

2016 China Team Selection Test, 5

Tags: geometry , cyclic quadrilateral

Refer to the diagram below. Let $ABCD$ be a cyclic quadrilateral with center $O$. Let the internal angle bisectors of $\angle A$ and $\angle C$ intersect at $I$ and let those of $\angle B$ and $\angle D$ intersect at $J$. Now extend $AB$ and $CD$ to intersect $IJ$ and $P$ and $R$ respectively and let $IJ$ intersect $BC$ and $DA$ at $Q$ and $S$ respectively. Let the midpoints of $PR$ and $QS$ be $M$ and $N$ respectively. Given that $O$ does not lie on the line $IJ$, show that $OM$ and $ON$ are perpendicular.

2019 Yasinsky Geometry Olympiad, p3

Tags: geometry , cyclic quadrilateral , angle chasing , angle

In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the diagonals $AC$ and $BD$.

2007 Indonesia TST, 1

Tags: circumcircle , cyclic quadrilateral , angle bisector , geometry

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.

2019 Yasinsky Geometry Olympiad, p3

Tags: geometry , bisects segment , cyclic quadrilateral

Let $ABCD$ be an inscribed quadrilateral whose diagonals are connected internally. are perpendicular to each other and intersect at the point $P$. Prove that the line connecting the midpoints of the opposite sides of the quadrilateral $ABCD$ bisects the lines $OP$ ($O$ is the center of the circle circumscribed around quadrilateral $ABCD$). (Alexander Dunyak)

2005 China Team Selection Test, 2

Tags: perimeter , cyclic quadrilateral , pythagorean theorem , geometry

Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.

2024 Brazil National Olympiad, 2

Tags: midpoint , triangle , congruent triangles , circumcircle , parallel , cyclic quadrilateral , geometry

Let $ ABC $ be a scalene triangle. Let $ E $ and $ F $ be the midpoints of sides $ AC $ and $ AB $, respectively, and let $ D $ be any point on segment $ BC $. The circumcircles of triangles $ BDF $ and $ CDE $ intersect line $ EF $ at points $ K \neq F $, and $ L \neq E $, respectively, and intersect at points $ X \neq D $. The point $ Y $ is on line $ DX $ such that $ AY $ is parallel to $ BC $. Prove that points $ K $, $ L $, $ X $, and $ Y $ lie on the same circle.

2018 CHKMO, 2

Tags: geometry , cyclic quadrilateral

Suppose $ABCD$ is a cyclic quadrilateral. Extend $DA$ and $DC$ to $P$ and $Q$ respectively such that $AP=BC$ and $CQ=AB$. Let $M$ be the midpoint of $PQ$. Show that $MA\perp MC$.

2014 China Team Selection Test, 1

Tags: circumcircle , geometric transformation , homothety , trigonometry , cyclic quadrilateral , geometry

Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly. Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent

2007 JBMO Shortlist, 2

Tags: trigonometry , geometry , incenter , function , circumcircle , cyclic quadrilateral , angle bisector

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2005 Postal Coaching, 16

Tags: geometry , trapezoid , circumcircle , parallelogram , cyclic quadrilateral

The diagonals AC and BD of a cyclic ABCD intersect at E. Let O be circumcentre of ABCD. If midpoints of AB, CD, OE are collinear prove that AD=BC. Bomb [color=red][Moderator edit: The problem is wrong. See also http://www.mathlinks.ro/Forum/viewtopic.php?t=53090 .][/color]

2024 USA TSTST, 4

Tags: geometry , circumcircle , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$. [i]Merlijn Staps[/i]