Olympiad problems

2006 IberoAmerican, 2

[color=darkred]The sides $AD$ and $CD$ of a tangent quadrilateral $ABCD$ touch the incircle $\varphi$ at $P$ and $Q,$ respectively. If $M$ is the midpoint of the chord $XY$ determined by $\varphi$ on the diagonal $BD,$ prove that $\angle AMP = \angle CMQ.$[/color]

1992 Turkey Team Selection Test, 1

Tags: cyclic quadrilateral , geometry

The feet of perpendiculars from the intersection point of the diagonals of cyclic quadrilateral $ABCD$ to the sides $AB,BC,CD,DA$ are $P,Q,R,S$, respectively. Prove $PQ+RS=QR+SP$.

2002 Tournament Of Towns, 5

Tags: geometry , projective geometry , cyclic quadrilateral

Two circles $\Gamma_1,\Gamma_2$ intersect at $A,B$. Through $B$ a straight line $\ell$ is drawn and $\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B)$. We are given $\ell_1\parallel AM$ is tangent to $\Gamma_1$ at $Q$. $QA\cap \Gamma_2=R\;(\neq A)$ and further $\ell_2$ is tangent to $\Gamma_2$ at $R$. Prove that: [list] [*]$\ell_2\parallel AK$ [*]$\ell,\ell_1,\ell_2$ have a common point.[/list]

2019 Nigerian Senior MO Round 3, 1

Tags: cyclic quadrilateral , geometry

Let the altitude from $A$ and $B$ of triangle $ABC$ meet the circumcircle of $ABC$ again at $D$ and $E$ respectively. Let $DE$ meet $AC$ and $BC$ at $P$ and $Q$ respectively. Show that $ABQP$ is cyclic

1992 National High School Mathematics League, 1

Tags: cyclic quadrilateral , geometry

$A_1A_2A_3A_4$ is cyclic quadrilateral of $\odot O$. $H_1,H_2,H_3,H_4$ are orthocentres of $\triangle A_2A_3A_4,\triangle A_3A_4A_1,\triangle A_4A_1A_2,\triangle A_1A_2A_3$. Prove that $H_1,H_2,H_3,H_4$ are concyclic, and determine its center.

2021 IMO Shortlist, G4

Tags: angle chasing , concurrency , 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.

2010 Contests, 2

Tags: circumcircle , cyclic quadrilateral , geometry , reflection , geometric transformation

Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$. Prove that $M_1$ lies on the segment $BH_1$.

Durer Math Competition CD Finals - geometry, 2011.C3

Tags: circles , cyclic quadrilateral , geometry

Given a circle with four circles that intersect in pairs as shown in the figure. The "internal" the points of intersection are $A, B, C$ and $D$, while the ‘outer’ points of intersection are $E, F, G$ and $H$. Prove that the quadrilateral $ABCD$ is cyclic if also the quadrilateral $EFGH$ is also cyclic. [img]https://cdn.artofproblemsolving.com/attachments/0/0/6a369c93e37eefd57775fd8586bdff393e1914.png[/img]

2015 Serbia National Math Olympiad, 1

Tags: geometry , cyclic quadrilateral

Consider circle inscribed quadriateral $ABCD$. Let $M,N,P,Q$ be midpoints of sides $DA,AB,BC,CD$.Let $E$ be the point of intersection of diagonals. Let $k1,k2$ be circles around $EMN$ and $EPQ$ . Let $F$ be point of intersection of $k1$ and $k2$ different from $E$. Prove that $EF$ is perpendicular to $AC$.

2022 Nigerian Senior MO Round 2, Problem 2

Tags: cyclic quadrilateral , concyclic , geometry

Let $G$ be the centroid of $\triangle ABC $ and let $D, E $ and $F$ be the midpoints of the line segments $BC, CA $ and $AB$ respectively. Suppose the circumcircle of $\triangle ABC $ meets $AD $ again at $X$, the circumcircle of $\triangle DEF $ meets $BE$ again at $Y$ and the circumcircle of $\triangle DEF $ meets $CF$ again at $Z$. Show that $G, X, Y $ and $Z$ are concyclic.

2023 Indonesia TST, 1

Tags: geometry , arc midpoint , cyclic quadrilateral

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

1950 Poland - Second Round, 3

Tags: angle bisector , geometry , cyclic quadrilateral

The diagonals of a quadrangle inscribed in a circle intersect at point $K$. The projections of the point $ K$ onto the subsequent sides of this quadrangle are points $M, N, P, Q$. Prove that these lines $KM$, $KN$, $KP$, $KQ$ are the angle bisectors of the quadrangle $MNPQ$.

2010 International Zhautykov Olympiad, 2

Tags: geometry , cyclic quadrilateral , circumcircle

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . 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 $MN$ .

2012 Online Math Open Problems, 27

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

Let $ABC$ be a triangle with circumcircle $\omega$. Let the bisector of $\angle ABC$ meet segment $AC$ at $D$ and circle $\omega$ at $M\ne B$. The circumcircle of $\triangle BDC$ meets line $AB$ at $E\ne B$, and $CE$ meets $\omega$ at $P\ne C$. The bisector of $\angle PMC$ meets segment $AC$ at $Q\ne C$. Given that $PQ = MC$, determine the degree measure of $\angle ABC$. [i]Ray Li.[/i]

2013 Peru IMO TST, 4

Tags: cyclic quadrilateral , geometry

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.

Croatia MO (HMO) - geometry, 2012.7

Tags: incircle , right triangle , geometry , concyclic , excircle , cyclic quadrilateral

Let the points $M$ and $N$ be the intersections of the inscribed circle of the right-angled triangle $ABC$, with sides $AB$ and $CA$ respectively , and points $P$ and $Q$ respectively be the intersections of the ex-scribed circles opposite to vertices $B$ and $C$ with direction $BC$. Prove that the quadrilateral $MNPQ$ is a cyclic if and only if the triangle $ABC$ is right-angled with a right angle at the vertex $A$.

2003 IberoAmerican, 2

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

Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.

2013 Saudi Arabia Pre-TST, 1.4

Tags: cyclic quadrilateral , centroid , geometry

$ABC$ is a triangle, $G$ its centroid and $A',B',C'$ the midpoints of its sides $BC,CA,AB$, respectively. Prove that if the quadrilateral $AC'GB'$ is cyclic then $AB \cdot CC' = AC \cdot BB'$:

2022 Switzerland Team Selection Test, 9

Tags: cyclic quadrilateral , angle chasing , concurrency , geometry

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.

2012 Korea Junior Math Olympiad, 5

Tags: equal segments , tangent , equal angles , geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscirbed in a circle $O$ ($AB> AD$), and let $E$ be a point on segment $AB$ such that $AE = AD$. Let $AC \cap DE = F$, and $DE \cap O = K(\ne D)$. The tangent to the circle passing through $C,F,E$ at $E$ hits $AK$ at $L$. Prove that $AL = AD$ if and only if $\angle KCE = \angle ALE$.

2011 Czech-Polish-Slovak Match, 2

Tags: geometry , symmetry , cyclic quadrilateral

In convex quadrilateral $ABCD$, let $M$ and $N$ denote the midpoints of sides $AD$ and $BC$, respectively. On sides $AB$ and $CD$ are points $K$ and $L$, respectively, such that $\angle MKA=\angle NLC$. Prove that if lines $BD$, $KM$, and $LN$ are concurrent, then \[ \angle KMN = \angle BDC\qquad\text{and}\qquad\angle LNM=\angle ABD.\]

2004 Cuba MO, 5

Tags: cyclic quadrilateral , geometry

Consider a circle $K$ and an inscribed quadrilateral $ABCD$, such that the diagonal $BD$ is not the diameter of the circle. Prove that the intersection of the lines tangent to $K$ through the points $B$ and $D$ lies on the line $AC$ if and only if $AB \cdot CD = AD \cdot BC$.

1999 Rioplatense Mathematical Olympiad, Level 3, 5

Tags: maximum , distance , semicircle , cyclic quadrilateral , geometry

The quadrilateral $ABCD$ is inscribed in a circle of radius $1$, so that $AB$ is a diameter of the circumference and $CD = 1$. A variable point $X$ moves along the semicircle determined by $AB$ that does not contain $C$ or $D$. Determine the position of $X$ for which the sum of the distances from $X$ to lines $BC, CD$ and $DA$ is maximum.

2017 Germany Team Selection Test, 3

Tags: geometry , cyclic quadrilateral , inversion , 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 $BDE$.

2018 Iranian Geometry Olympiad, 5

Tags: geometry , concyclic , incenter , cyclic quadrilateral

$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$. Another circle passing through $C,D$ is tangent to $AB$ at point $F$. Point $G$ is the intersection point of $AE,DF$, and point $H$ is the intersection point of $BE$, $CF$. Prove that the incenters of triangles $AGF$, $BHF$, $CHE$, $DGE$ lie on a circle. Proposed by Le Viet An (Vietnam)