Olympiad problems

1996 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6

A square $ ABCD$ is inscribed in a circle. Let $ \alpha \equal{} \angle DAB, \beta \equal{} \angle BDA,$ and $ \gamma \equal{} \angle CDB$. Then $ \angle DBC$ equals A. $ \alpha \minus{} \beta$ B. $ \alpha \minus{} \gamma$ C. $ 90^\circ \minus{} \alpha \plus{} \beta$ D. $ 90^\circ \minus{} \alpha \plus{} \gamma$ E. $ 180^\circ \minus{} \alpha \minus{} \gamma$

2015 Canadian Mathematical Olympiad Qualification, 4

Tags: cyclic quadrilateral , geometry

Given an acute-angled triangle $ABC$ whose altitudes from $B$ and $C$ intersect at $H$, let $P$ be any point on side $BC$ and $X, Y$ be points on $AB, AC$, respectively, such that $PB = PX$ and $PC = PY$. Prove that the points $A, H, X, Y$ lie on a common circle.

2014 Greece National Olympiad, 4

Tags: geometry , projective geometry , cyclic quadrilateral

We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.

1994 Poland - First Round, 3

Tags: geometry , cyclic quadrilateral

A quadrilateral with sides $a,b,c,d$ is inscribed in a circle of radius $R$. Prove that if $a^2+b^2+c^2+d^2=8R^2$, then either one of the angles of the quadrilateral is right or the diagonals of the quadrilateral are perpendicular.

1998 Romania National Olympiad, 2

Tags: cyclic quadrilateral , inequalities , geometry

Let $ABCD$ be a cyclic quadrilateral. Show that $\vert \overline{AC} - \overline{BD} \vert \le \vert \overline{AB}-\overline{CD} \vert$ and determine when does equality hold.

Brazil L2 Finals (OBM) - geometry, 2023.2

Tags: circumcenter , isogonal lines , similar triangles , orthocenter , midpoint , cyclic quadrilateral , geometry

Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.

2005 MOP Homework, 5

Tags: geometry , search , inequalities , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$. Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$. [color=#FF0000]Moderator says: Use the search before posting contest problems [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=530783[/url][/color]

Kyiv City MO Seniors 2003+ geometry, 2012.10.4

Tags: concyclic , geometry , circumcircle , cyclic quadrilateral

The triangle $ABC$ with $AB> AC$ is inscribed in a circle, the angle bisector of $\angle BAC$ intersects the side $BC$ of the triangle at the point $K$, and the circumscribed circle at the point $M$. The midline of $\Delta ABC$, which is parallel to the side $AB$, intersects $AM$ at the point $O$, the line $CO$ intersects the line $AB$ at the point $N$. Prove that a circle can be circumscribed around the quadrilateral $BNKM$. (Nagel Igor)

2006 Sharygin Geometry Olympiad, 10.6

Tags: tangential quadrilateral , construction , circumcenter , geometry , incenter , cyclic quadrilateral

A quadrangle was drawn on the board, that you can inscribe and circumscribe a circle. Marked are the centers of these circles and the intersection point of the lines connecting the midpoints of the opposite sides, after which the quadrangle itself was erased. Restore it with a compass and ruler.

2022 Macedonian Mathematical Olympiad, Problem 2

Tags: circumcircle , geometry , cyclic quadrilateral , reflection

Let $ABCD$ be cyclic quadrilateral and $E$ the midpoint of $AC$. The circumcircle of $\triangle CDE$ intersect the side $BC$ at $F$, which is different from $C$. If $B'$ is the reflection of $B$ across $F$, prove that $EF$ is tangent to the circumcircle of $\triangle B'DF$. [i]Proposed by Nikola Velov[/i]

2004 Junior Balkan Team Selection Tests - Romania, 2

Tags: equilateral , centroid , midpoint , geometry , cyclic quadrilateral

Let $M,N, P$ be the midpoints of the sides $BC,CA,AB$ of the triangle $ABC$, respectively, and let $G$ be the centroid of the triangle. Prove that if $BMGP$ is cyclic and $2BN = \sqrt3 AB$ , then triangle $ABC$ is equilateral.

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.

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: orthocenter , concyclic , cyclic quadrilateral , circumcircle , parallel , geometry

Let $ABC$ be a triangle in which $AB < AC, D$ is the foot of the altitude from $A, H$ is the orthocenter, $O$ is the circumcenter, $M$ is the midpoint of the side $BC, A'$ is the reflection of $A$ across $O$, and $S$ is the intersection of the tangents at $B$ and $C$ to the circumcircle. The tangent at $A'$ to the circumcircle intersects $SC$ and $SB$ at $X$ and $Y$ , respectively. If $M,S,X,Y$ are concyclic, prove that lines $OD$ and $SA'$ are parallel.

2010 Contests, 3

Tags: cyclic quadrilateral , angle chasing , geometry , circumcircle , radical axis

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]

2022 Austrian MO National Competition, 2

Tags: geometry , cyclic quadrilateral , equal segments

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]

2015 Saudi Arabia Pre-TST, 1.1

Tags: symmetric , concyclic , cyclic quadrilateral , geometry

Let $ABC$ be a triangle and $D$ a point on the side $BC$. Point $E$ is the symmetric of $D$ with respect to $AB$. Point $F$ is the symmetric of $E$ with respect to $AC$. Point $P$ is the intersection of line $DF$ with line $AC$. Prove that the quadrilateral $AEDP$ is cyclic. (Malik Talbi)

Ukraine Correspondence MO - geometry, 2013.9

Tags: geometry , equal circles , cyclic quadrilateral

Let $E$ be the point of intersection of the diagonals of the cyclic quadrilateral $ABCD$, and let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$, respectively. Prove that the radii of the circles circumscribed around the triangles $KLE$ and $MNE$ are equal.

2023 India Regional Mathematical Olympiad, 6

Tags: circumcircle , geometry , cyclic quadrilateral

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $P$. The point $Q$ is chosen on the segment $BC$ so that $PQ$ is perpendicular to $AC$. Prove that the line joining the centres of the circumcircles of triangles $APD$ and $BQD$ is parallel to $AD$.

2022 Korea Winter Program Practice Test, 5

Tags: concurrency , cyclic quadrilateral , geometry

Let $ABDC$ be a cyclic quadrilateral inscribed in a circle $\Omega$. $AD$ meets $BC$ at $P$, and $\Omega$ meets lines passing $A$ and parallel to $DB$, $DC$ at $E$, $F$, respectively. $X$ is a point on $\Omega$ such that $PA=PX$. Prove that the lines $BE$, $CF$, and $DX$ are concurrent.

1999 Poland - Second Round, 3

Tags: area , cyclic quadrilateral , ratio , area of a triangle , geometry

Let $ABCD$ be a cyclic quadrilateral and let $E$ and $F$ be the points on the sides $AB$ and $CD$ respectively such that $AE : EB = CF : FD$. Point $P$ on the segment EF satsfies $EP : PF = AB : CD$. Prove that the ratio of the areas of $\vartriangle APD$ and $\vartriangle BPC$ does not depend on the choice of $E$ and $F$.

2021 Centroamerican and Caribbean Math Olympiad, 6

Tags: cyclic quadrilateral , geometry

Let $ABC$ be a triangle with $AB<AC$ and let $M$ be the midpoint of $AC$. A point $P$ (other than $B$) is chosen on the segment $BC$ in such a way that $AB=AP$. Let $D$ be the intersection of $AC$ with the circumcircle of $\bigtriangleup ABP$ distinct from $A$, and $E$ be the intersection of $PM$ with the circumcircle of $\bigtriangleup ABP$ distinct from $P$. Let $K$ be the intersection of lines $AP$ and $DE$. Let $F$ be a point on $BC$ (other than $P$) such that $KP=KF$. Show that $C,\ D,\ E$ and $F$ lie on the same circle.

2019 Iran MO (3rd Round), 1

Tags: cyclic quadrilateral , geometry

Given a cyclic quadrilateral $ABCD$. There is a point $P$ on side $BC$ such that $\angle PAB=\angle PDC=90^\circ$. The medians of vertexes $A$ and $D$ in triangles $PAB$ and $PDC$ meet at $K$ and the bisectors of $\angle PAB$ and $\angle PDC$ meet at $L$. Prove that $KL\perp BC$.

2016 China Team Selection Test, 3

Tags: cyclic quadrilateral , incenter , geometry

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 $AC$.

2016 Bulgaria JBMO TST, 1

Tags: geometric inequality , angle , geometry , cyclic quadrilateral

The quadrilateral $ABCD$, in which $\angle BAC < \angle DCB$ , is inscribed in a circle $c$, with center $O$. If $\angle BOD = \angle ADC = \alpha$. Find out which values of $\alpha$ the inequality $AB <AD + CD$ occurs.

2016 German National Olympiad, 5

Tags: angle , cyclic quadrilateral , geometry , trigonometry

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.