Olympiad problems

1990 APMO, 1

Given triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $AC$, $AB$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle BAC$, how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?

2009 International Zhautykov Olympiad, 2

Tags: cyclic quadrilateral , similar triangles , geometry

Given a quadrilateral $ ABCD$ with $ \angle B\equal{}\angle D\equal{}90^{\circ}$. Point $ M$ is chosen on segment $ AB$ so taht $ AD\equal{}AM$. Rays $ DM$ and $ CB$ intersect at point $ N$. Points $ H$ and $ K$ are feet of perpendiculars from points $ D$ and $ C$ to lines $ AC$ and $ AN$, respectively. Prove that $ \angle MHN\equal{}\angle MCK$.

1989 China Team Selection Test, 4

Tags: geometry , circumcircle , geometric transformation , rotation , congruent triangles , cyclic quadrilateral

Given triangle $ABC$, squares $ABEF, BCGH, CAIJ$ are constructed externally on side $AB, BC, CA$, respectively. Let $AH \cap BJ = P_1$, $BJ \cap CF = Q_1$, $CF \cap AH = R_1$, $AG \cap CE = P_2$, $BI \cap AG = Q_2$, $CE \cap BI = R_2$. Prove that triangle $P_1 Q_1 R_1$ is congruent to triangle $P_2 Q_2 R_2$.

2022 Saudi Arabia IMO TST, 2

Tags: geometry , cyclic quadrilateral , concurrency , angle chasing

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.

Geometry Mathley 2011-12, 7.4

Tags: tangent , parallel , cyclic quadrilateral , geometry , tangent circle

Let $ABCD$ be a quadrilateral inscribed in the circle $(O)$. Let $(K)$ be an arbitrary circle passing through $B,C$. Circle $(O_1)$ tangent to $AB,AC$ and is internally tangent to $(K)$. Circle $(O_2)$ touches $DB,DC$ and is internally tangent to $(K)$. Prove that one of the two external common tangents of $(O_1)$ and $(O_2)$ is parallel to $AD$. Trần Quang Hùng

2014 Baltic Way, 5

Tags: geometry , trigonometry , cyclic quadrilateral , algebra

Given positive real numbers $a, b, c, d$ that satisfy equalities \[a^2 + d^2 - ad = b^2 + c^2 + bc \ \ \text{and} \ \ a^2 + b^2 = c^2 + d^2\] find all possible values of the expression $\frac{ab+cd}{ad+bc}.$

2015 Saudi Arabia Pre-TST, 1.1

Tags: geometry , symmetric , cyclic quadrilateral , concyclic

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)

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 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 $AB$.

2020 China Northern MO, BP4

Tags: incenter , orthocenter , circumcenter , cyclic quadrilateral , spiral similarity , geometry

In $\triangle ABC$, $\angle BAC = 60^{\circ}$, point $D$ lies on side $BC$, $O_1$ and $O_2$ are the centers of the circumcircles of $\triangle ABD$ and $\triangle ACD$, respectively. Lines $BO_1$ and $CO_2$ intersect at point $P$. If $I$ is the incenter of $\triangle ABC$ and $H$ is the orthocenter of $\triangle PBC$, then prove that the four points $B,C,I,H$ are on the same circle.

2022 Taiwan TST Round 3, 1

Tags: geometry , cyclic quadrilateral , concurrency , angle chasing

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.

2005 All-Russian Olympiad Regional Round, 9.4

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

9.4, 10.3 Let $I$ be an incenter of $ABC$ ($AB<BC$), $M$ is a midpoint of $AC$, $N$ is a midpoint of circumcircle's arc $ABC$. Prove that $\angle IMA=\angle INB$. ([i]A. Badzyan[/i])

2010 Contests, 2

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

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 $BH_1$.

Swiss NMO - geometry, 2020.7

Tags: geometry , trapezoid , angle bisector , cyclic quadrilateral

Let $ABCD$ be an isosceles trapezoid with bases $AD> BC$. Let $X$ be the intersection of the bisectors of $\angle BAC$ and $BC$. Let $E$ be the intersection of$ DB$ with the parallel to the bisector of $\angle CBD$ through $X$ and let $F$ be the intersection of $DC$ with the parallel to the bisector of $\angle DCB$ through $X$. Show that quadrilateral $AEFD$ is cyclic.

2023 Estonia Team Selection Test, 3

Tags: arc midpoint , cyclic quadrilateral , geometry

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.

2018 Saudi Arabia IMO TST, 3

Tags: geometry , angle bisector , cyclic quadrilateral , parallel , equal segments , perpendicular

Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.

2013 Brazil Team Selection Test, 2

Tags: geometry , acute , cyclic quadrilateral , angle

Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.

1972 IMO Shortlist, 10

Tags: geometry , trapezoid , dissection , cyclic quadrilateral

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

2022 Korea Winter Program Practice Test, 1

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Omega$ such that $AB<CD$. Suppose that $AC$ meets $BD$ at $E$, $AD$ meets $BC$ at $F$, and $\Omega$ meets $(FAE)$, $(FBE)$ at $X$, $Y$, respectively. Prove that if $XY$ is diameter of $\Omega$, then $XY$ is perpendicular to $EF$.

2005 Sharygin Geometry Olympiad, 9.1

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

2015 Federal Competition For Advanced Students, P2, 2

Tags: geometry , cyclic quadrilateral

We are given a triangle $ABC$. Let $M$ be the mid-point of its side $AB$. Let $P$ be an interior point of the triangle. We let $Q$ denote the point symmetric to $P$ with respect to $M$. Furthermore, let $D$ and $E$ be the common points of $AP$ and $BP$ with sides $BC$ and $AC$, respectively. Prove that points $A$, $B$, $D$, and $E$ lie on a common circle if and only if $\angle ACP = \angle QCB$ holds. (Karl Czakler)

2004 Postal Coaching, 1

Tags: circumcircle , cyclic quadrilateral , geometry

Let $ABC$ and $DEF$ be two triangles such that $A+ D = 120^{\circ}$ and $B+E = 120^{\circ}$. Suppose they have the same circumradius. Prove that they have the same 'Fermat length'.

2000 Junior Balkan MO, 3

Tags: symmetry , circumcircle , angle bisector , projective geometry , cyclic quadrilateral , geometry

A half-circle of diameter $EF$ is placed on the side $BC$ of a triangle $ABC$ and it is tangent to the sides $AB$ and $AC$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$. [i]Albania[/i]

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 $BE = BF$.

2023 Brazil Cono Sur TST, 4

Tags: cyclic quadrilateral , power of a point

The diagonals of a cyclic quadrilateral $ABCD$ meet at $P$. Let $K$ and $L$ be points on the segments $CP$ and $DP$ such that the circumcircle of $PKL$ is tangent to $CD$ at $M$. Let $X$ and $Y$ be points on the segments $AP$ and $BP$ such that $AX=CK$ and $BY=DL$. Points $Z$ and $W$ are the midpoints of $PK$ and $PL$. Prove that if $C,D,X$ and $Y$ are concyclic, then $\angle MZP = \angle MWP$.