Olympiad problems

2025 Bangladesh Mathematical Olympiad, P3

Let $ABC$ be a given triangle with circumcenter $O$ and orthocenter $H$. Let $D, E$ and $F$ be the feet of the perpendiculars from $A, B$ and $C$ to the opposite sides, respectively. Let $A'$ be the reflection of $A$ with respect to $EF$. Prove that $HOA'D$ is a cyclic quadrilateral. [i]Proposed by Imad Uddin Ahmad Hasin[/i]

2016 China Western Mathematical Olympiad, 7

Tags: cyclic quadrilateral , geometry

$ABCD$ is a cyclic quadrilateral, and $\angle BAC = \angle DAC$. $\astrosun I_1$ and $\astrosun I_2$ are the incircles of $\triangle ABD$ and $\triangle ADC$ respectively. Prove that one of the common external tangents of $\astrosun I_1$ and $\astrosun I_2$ is parallel to $BD$

2007 All-Russian Olympiad, 4

Tags: cyclic quadrilateral , asymptote , circumcircle , geometry , angle bisector

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

2014 Sharygin Geometry Olympiad, 1

Tags: geometric inequality , geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$. (V. Yasinsky)

2012 Indonesia TST, 3

Tags: incenter , projective geometry , circumcircle , geometry , cyclic quadrilateral

Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.

2022 Latvia Baltic Way TST, P9

Tags: geometry , circumcircle , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let the lines $AB$ and $CD$ intersect at $P$, and the lines $AD$ and $BC$ intersect at $Q$. Let then the circumcircle of the triangle $\triangle APQ$ intersect $\Omega$ at $R \neq A$. Prove that the line $CR$ goes through the midpoint of the segment $PQ$.

2008 Princeton University Math Competition, 5

Tags: cyclic quadrilateral

Quadrilateral $ABCD$ has both an inscribed and a circumscribed circle and sidelengths $BC = 4, CD = 5, DA = 6$. Find the area of $ABCD$.

2007 China Team Selection Test, 2

Tags: incenter , circumcircle , euler , projective geometry , symmetry , geometry , cyclic quadrilateral

Let $ ABCD$ be the inscribed quadrilateral with the circumcircle $ \omega$.Let $ \zeta$ be another circle that internally tangent to $ \omega$ and to the lines $ BC$ and $ AD$ at points $ M,N$ respectively.Let $ I_1,I_2$ be the incenters of the $ \triangle ABC$ and $ \triangle ABD$.Prove that $ M,I_1,I_2,N$ are collinear.

Indonesia MO Shortlist - geometry, g3

Tags: geometry , equal segments , cyclic quadrilateral

Given a quadrilateral $ABCD$ inscribed in circle $\Gamma$.From a point P outside $\Gamma$, draw tangents $PA$ and $PB$ with $A$ and $B$ as touspoints. The line $PC$ intersects $\Gamma$ at point $D$. Draw a line through $B$ parallel to $PA$, this line intersects $AC$ and $AD$ at points $E$ and $F$ respectively. Prove that $BE = BF$.

2008 Estonia Team Selection Test, 2

Tags: geometry , angle bisector , cyclic , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

2008 Serbia National Math Olympiad, 6

Tags: trigonometry , inequalities , geometry , cyclic quadrilateral

In a convex pentagon $ ABCDE$, let $ \angle EAB \equal{} \angle ABC \equal{} 120^{\circ}$, $ \angle ADB \equal{} 30^{\circ}$ and $ \angle CDE \equal{} 60^{\circ}$. Let $ AB \equal{} 1$. Prove that the area of the pentagon is less than $ \sqrt {3}$.

2019 China Girls Math Olympiad, 7

Tags: cyclic quadrilateral , geometry

Let $DFGE$ be a cyclic quadrilateral. Line $DF$ intersects $EG$ at $C,$ and line $FE$ intersects $DG$ at $H.$ $J$ is the midpoint of $FG.$ The line $\ell$ is the reflection of the line $DE$ in $CH,$ and it intersects line $GF$ at $I.$ Prove that $C,J,H,I$ are concyclic.

2010 Balkan MO Shortlist, G2

Tags: midpoint , cyclic , geometric inequality , circles , area , cyclic quadrilateral , geometry

Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle

2010 Dutch IMO TST, 4

Tags: cyclic quadrilateral , midpoint , equal angles , geometry

Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.

2017 Romanian Masters In Mathematics, 6

Tags: cyclic quadrilateral , geometry

Let $ABCD$ be any convex quadrilateral and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrilaterals, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.

2014 USA Team Selection Test, 2

Tags: analytic geometry , area , geometry , orthocenter , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, and let $E$, $F$, $G$, and $H$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. Let $W$, $X$, $Y$ and $Z$ be the orthocenters of triangles $AHE$, $BEF$, $CFG$ and $DGH$, respectively. Prove that the quadrilaterals $ABCD$ and $WXYZ$ have the same area.

2021 Centroamerican and Caribbean Math Olympiad, 2

Tags: circumcircle , cyclic quadrilateral , geometry

Let $ABC$ be a triangle and let $\Gamma$ be its circumcircle. Let $D$ be a point on $AB$ such that $CD$ is parallel to the line tangent to $\Gamma$ at $A$. Let $E$ be the intersection of $CD$ with $\Gamma$ distinct from $C$, and $F$ the intersection of $BC$ with the circumcircle of $\bigtriangleup ADC$ distinct from $C$. Finally, let $G$ be the intersection of the line $AB$ and the internal bisector of $\angle DCF$. Show that $E,\ G,\ F$ and $C$ lie on the same circle.

2022 Brazil Team Selection Test, 2

Tags: concurrency , angle chasing , geometry , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2011 Grand Duchy of Lithuania, 4

Tags: geometry , orthocenter , equal segments , cyclic quadrilateral

In the cyclic quadrilateral $ABCD$ with $AB = AD$, points $M$ and $N$ lie on the sides $CD$ and $BC$ respectively so that $MN = BN + DM$. Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$.

2020 Sharygin Geometry Olympiad, 2

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral. A circle passing through $A$ and $B$ meets $AC$ and $BD$ at points $E$ and $F$ respectively. The lines $AF$ and $BC$ meet at point $P$, and the lines $BE$ and $AD$ meet at point $Q$. Prove that $PQ$ is parallel to $CD$.

2023 Greece JBMO TST, 2

Tags: geometry , cyclic quadrilateral

Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.

2017 CMIMC Geometry, 9

Tags: geometry , cyclic quadrilateral

Let $\triangle ABC$ be an acute triangle with circumcenter $O$, and let $Q\neq A$ denote the point on $\odot (ABC)$ for which $AQ\perp BC$. The circumcircle of $\triangle BOC$ intersects lines $AC$ and $AB$ for the second time at $D$ and $E$ respectively. Suppose that $AQ$, $BC$, and $DE$ are concurrent. If $OD=3$ and $OE=7$, compute $AQ$.

2019 Czech-Polish-Slovak Junior Match, 3

Tags: geometry , circumcircle , cyclic quadrilateral

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals, such that $\angle BAC = \angle ADB$, $\angle CBD = \angle DCA$, $AB = 15$, $CD = 8$. Show that $ABCD$ is cyclic and find the distance between its circumcenter and the intersection point of its diagonals.

2005 Sharygin Geometry Olympiad, 9.1

Tags: perpendicular , equal angles , diagonal , cyclic quadrilateral , geometry

The quadrangle $ABCD$ is inscribed in a circle whose center $O$ lies inside it. Prove that if $\angle BAO = \angle DAC$, then the diagonals of the quadrilateral are perpendicular.

2023 NMTC Junior, P8

Tags: geometry , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral. The midpoints of the diagonals $AC$ and $BD$ are respectively $P$ and $Q$. If $BD$ bisects $\angle AQC$, the prove that $AC$ will bisect $\angle BPD$.