Olympiad problems

2016 Puerto Rico Team Selection Test, 3

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 $PS = PT = PU = PV$.

2013 Oral Moscow Geometry Olympiad, 1

Tags: incenter , cyclic quadrilateral , circumcircle , geometry

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $O$. The circumscribed circles of triangles $AOB$ and $COD$ intersect at point $M$ on the side $AD$. Prove that the point $O$ is the center of the inscribed circle of the triangle $BMC$.

2021-IMOC, G1

Tags: geometry , altitude , antipode , cyclic quadrilateral

Let $\overline{BE}$ and $\overline{CF}$ be altitudes of triangle $ABC$, and let $D$ be the antipodal point of $A$ on the circumcircle of $ABC$. The lines $\overleftrightarrow{DE}$ and $\overleftrightarrow{DF}$ intersect $\odot(ABC)$ again at $Y$ and $Z$, respectively. Show that $\overleftrightarrow{YZ}$, $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect at a point.

2017 Taiwan TST Round 1, 2

Tags: geometry , inversion , cyclic quadrilateral , reflection , incenter

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2015 Turkey MO (2nd round), 5

Tags: geometry , cyclic quadrilateral

In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.

2011 Balkan MO, 1

Tags: trapezoid , similar triangles , geometry , cyclic quadrilateral , trigonometry

Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.

1998 India National Olympiad, 4

Tags: rectangle , geometric inequality , trigonometry , cyclic quadrilateral , geometry

Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.

2017 Romanian Master of 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.

2007 Czech and Slovak Olympiad III A, 2

Tags: geometry , cyclic quadrilateral , incenter

In a cyclic quadrilateral $ABCD$, let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$.Prove that triangle $LMR$ is isosceles.

2003 Turkey Junior National Olympiad, 1

Tags: trig identities , quadratic , law of cosines , geometry , trigonometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$, $|BD|=6$, and $|AD|\cdot|CE|=|DC|\cdot|AE|$, find the area of the quadrilateral $ABCD$.

2018 Romania Team Selection Tests, 1

Tags: incenter , geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral and let its diagonals $AC$ and $BD$ cross at $X$. Let $I$ be the incenter of $XBC$, and let $J$ be the center of the circle tangent to the side $BC$ and the extensions of sides $AB$ and $DC$ beyond $B$ and $C$. Prove that the line $IJ$ bisects the arc $BC$ of circle $ABCD$, not containing the vertices $A$ and $D$ of the quadrilateral.

Croatia MO (HMO) - geometry, 2012.3

Tags: equal segments , geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral such that $|AD| =|BD|$ and let $M$ be the intersection of its diagonals. Furthermore, let $N$ be the second intersection of the diagonal $AC$ with the circle passing through points $B, M$ and the center of the circle inscribed in triangle $BCM$. Prove that $AN \cdot NC = CD \cdot BN$

2007 Thailand Mathematical Olympiad, 2

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral so that arcs $AB$ and $BC$ are equal. Given that $AD = 6, BD = 4$ and $CD = 1$, compute $AB$.

2022 Federal Competition For Advanced Students, P1, 2

Tags: equal segments , cyclic quadrilateral , geometry

The points $A, B, C, D$ lie in this order on a circle with center $O$. Furthermore, the straight lines $AC$ and $BD$ should be perpendicular to each other. The base of the perpendicular from $O$ on $AB$ is $F$. Prove $CD = 2 OF$. [i](Karl Czakler)[/i]

2022 Germany Team Selection Test, 2

Tags: geometry , angle chasing , concurrency , 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.

2004 France Team Selection Test, 2

Tags: incenter , inequalities , cyclic quadrilateral , inradius , perimeter , trigonometry , geometry

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2007 IMO Shortlist, 2

Tags: reflection , circumcircle , geometry , mixtilinear incircle , cyclic quadrilateral

Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$. Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$. [i]Author: Farzan Barekat, Canada[/i]

1990 Canada National Olympiad, 3

Tags: incenter , geometry , angle bisector , cyclic quadrilateral

The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.

2011 Sharygin Geometry Olympiad, 5

Tags: equal segments , cyclic quadrilateral , geometry

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.

2001 Baltic Way, 9

Tags: circumcircle , rhombus , parallelogram , hyperbola , geometry , cyclic quadrilateral , conic

Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.

2003 IMO Shortlist, 1

Tags: quadrilateral , concurrency , geometry , angle bisector , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2021 Macedonian Balkan MO TST, Problem 1

Tags: geometry , cyclic quadrilateral

Let $ABC$ be an acute triangle. Let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively and let $H$ be the orthocenter of $\triangle ABC$. Let $X$ be an arbitrary point on the circumcircle of $\triangle DEF$ and let the circumcircles of $\triangle EHX$ and $\triangle FHX$ intersect the second time the lines $CF$ and $BE$ second at $Y$ and $Z$, respectively. Prove that the line $YZ$ passes through the midpoint of $BC$.

2015 Belarus Team Selection Test, 2

Tags: equal segments , geometry , circumcenter , cyclic quadrilateral

Given a cyclic $ABCD$ with $AB=AD$. Points $M$ and $N$ are marked on the sides $CD$ and $BC$, respectively, so that $DM+BN=MN$. Prove that the circumcenter of the triangle $AMN$ belongs to the segment $AC$. N.Sedrakian

2005 Portugal MO, 5

Tags: geometry , tangential , cyclic quadrilateral

Considers a quadrilateral $[ABCD]$ that has an inscribed circle and a circumscribed circle. The sides $[AD]$ and $[BC]$ are tangent to the circle inscribed at points $E$ and $F$, respectively. Prove that $AE \cdot F C = BF \cdot ED$. [img]https://1.bp.blogspot.com/-6o1fFTdZ69E/X4XMo98ndAI/AAAAAAAAMno/7FXiJnWzJgcfSn-qSRoEAFyE8VgxmeBjwCLcBGAsYHQ/s0/2005%2BPortugal%2Bp5.png[/img]

2004 Korea Junior Math Olympiad, 4

Tags: rectangle , excircle , cyclic quadrilateral , geometry

$ABCD$ is a cyclic quadrilateral inscribed in circle $O$. Let $O_1$ be the $A$-excenter of $ABC$ and $O_2$ the $A$-excenter of $ABD$. Show that $A, B, O_1, O_2$ is concyclic, and $O$ passes through the center of $(ABO_1O_2)$. Recall that for concyclic $X, Y, Z, W$, the notation $(XYZW)$ denotes the circumcircle of $XYZW$.