Olympiad problems

2019 Dutch IMO TST, 4

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic

2022 Irish Math Olympiad, 7

Tags: geometry , circles , cyclic quadrilateral

7. The four Vertices of a quadrilateral [i]ABCD[/i] lie on the circle with diameter [i]AB[/i]. The diagonals of [i]ABCD[/i] intersect at [i]E[/i], and the lines [i]AD[/i] and [i]BC[/i] intersect at [i]F[/i]. Line [i]FE[/i] meets [i]AB[/i] at [i]K[/i] and line [i]DK[/i] meets the circle again at [i]L[/i]. Prove that [i]CL[/i] is perpendicular to [i]AB[/i].

2024 Baltic Way, 11

Tags: circumcircle , cyclic quadrilateral , geometry

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O$ and with $AC$ perpendicular to $BD$. Points $X$ and $Y$ lie on the circumcircle of the triangle $BOD$ such that $\angle AXO=\angle CYO=90^{\circ}$. Let $M$ be the midpoint of $AC$. Prove that $BD$ is tangent to the circumcircle of the triangle $MXY$.

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.

2010 Middle European Mathematical Olympiad, 10

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

Let $A$, $B$, $C$, $D$, $E$ be points such that $ABCD$ is a cyclic quadrilateral and $ABDE$ is a parallelogram. The diagonals $AC$ and $BD$ intersect at $S$ and the rays $AB$ and $DC$ intersect at $F$. Prove that $\sphericalangle{AFS}=\sphericalangle{ECD}$. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 6)[/i]

2020 Kosovo Team Selection Test, 3

Tags: geometry , angle bisector , cyclic quadrilateral , harmonic range

Let $ABCD$ be a cyclic quadrilateral with center $O$ such that $BD$ bisects $AC.$ Suppose that the angle bisector of $\angle ABC$ intersects the angle bisector of $\angle ADC$ at a single point $X$ different than $B$ and $D.$ Prove that the line passing through the circumcenters of triangles $XAC$ and $XBD$ bisects the segment $OX.$ [i]Proposed by Viktor Ahmeti and Leart Ajvazaj, Kosovo[/i]

2003 Korea - Final Round, 1

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

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 $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2013 Peru IMO TST, 4

Tags: geometry , cyclic quadrilateral

Let $A$ be a point outside of a circumference $\omega$. Through $A$, two lines are drawn that intersect $\omega$, the first one cuts $\omega$ at $B$ and $C$, while the other one cuts $\omega$ at $D$ and $E$ ($D$ is between $A$ and $E$). The line that passes through $D$ and is parallel to $BC$ intersects $\omega$ at point $F \neq D$, and the line $AF$ intersects $\omega$ at $T \neq F$. Let $M$ be the intersection point of lines $BC$ and $ET$, $N$ the point symmetrical to $A$ with respect to $M$, and $K$ be the midpoint of $BC$. Prove that the quadrilateral $DEKN$ is cyclic.

2016 Iran MO (3rd Round), 2

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

Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively. Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. Prove that $XYII_a$ is cyclic.

OMMC POTM, 2023 4

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $CDX$ and $ABY$ meet line $XY$ again at $P$ and $Q$ respectively. Show that $OP=OQ$. [i]Proposed by Evan Chang (squareman), USA[/i]

I Soros Olympiad 1994-95 (Rus + Ukr), 10.9

Tags: combinatorics , geometry , combinatorial geometry , cyclic quadrilateral

Prove that for all natural $n\ge 6 000$ any convex $1994$-gon can be cut into $n$ such quadrilaterals thata circle can be circumscribed around each of them

1972 IMO, 2

Tags: geometry , trapezoid , dissection , cyclic quadrilateral

Given $n>4$, prove that every cyclic quadrilateral can be dissected into $n$ cyclic quadrilaterals.

Oliforum Contest III 2012, 5

Tags: geometry , cyclic quadrilateral , orthogonal

Consider a cyclic quadrilateral $ABCD$ and define points $X = AB \cap CD$, $Y = AD \cap BC$, and suppose that there exists a circle with center $Z$ inscribed in $ABCD$. Show that the $Z$ belongs to the circle with diameter $XY$ , which is orthogonal to circumcircle of $ABCD$.

Geometry Mathley 2011-12, 16.1

Tags: euler line , geometry , cyclic quadrilateral , concurrency

Let $ABCD$ be a cyclic quadrilateral with two diagonals intersect at $E$. Let $ M$, $N$, $P$, $Q$ be the reflections of $ E $ in midpoints of $AB$, $BC$, $CD$, $DA$ respectively. Prove that the Euler lines of $ \triangle MAB$, $\triangle NBC$, $\triangle PCD,$ $\triangle QDA$ are concurrent. Trần Quang Hùng

2013 India IMO Training Camp, 2

Tags: circumcircle , trigonometry , cyclic quadrilateral , perpendicular bisector , geometry

Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.

2021 Nigerian Senior MO Round 2, 5

Tags: cyclic quadrilateral , circumcircle , angle bisector , geometry

let $ABCD$ be a cyclic quadrilateral with $E$,an interior point such that $AB=AD=AE=BC$. Let $DE$ meet the circumcircle of $BEC$ again at $F$. Suppose a common tangent to the circumcircle of $BEC$ and $DEC$ touch the circles at $F$ and $G$ respectively. Show that $GE$ is the external angle bisector of angle $BEF$

2017 Germany Team Selection Test, 3

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

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 $BDE$.

1993 India National Olympiad, 1

Tags: geometry , circumcircle , cyclic quadrilateral , power of a point , radical axis

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ intersect at $P$. Let $O$ be the circumcenter of triangle $APB$ and $H$ be the orthocenter of triangle $CPD$. Show that the points $H,P,O$ are collinear.

1986 China Team Selection Test, 1

Tags: incenter , rectangle , cyclic quadrilateral , angle chasing , geometry

If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.

Kyiv City MO 1984-93 - geometry, 1990.10.3

Tags: geometry , construction , cyclic quadrilateral

Construct a quadrilateral with three sides $1$, $4$ and $3$ so that a circle could be circumscribed around it.

2003 All-Russian Olympiad, 2

Tags: circumcircle , cyclic quadrilateral , geometric transformation , spiral similarity , geometry

The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.

2008 Indonesia TST, 1

Tags: geometry , incenter , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, and angle bisectors of $\angle BAD$ and $\angle BCD$ meet at point $I$. Show that if $\angle BIC = \angle IDC$, then $I$ is the incenter of triangle $ABD$.

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]

2016 China Team Selection Test, 6

Tags: geometry , cyclic quadrilateral , harmonic division

The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.

2011 Sharygin Geometry Olympiad, 20

Tags: midpoint , diagonal , geometry , cyclic quadrilateral , tangential quadrilateral

Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$. [i]Nikolai Beluhov and Aleksey Zaslavsky[/i]