Olympiad problems

2009 International Zhautykov Olympiad, 2

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$.

2018 Puerto Rico Team Selection Test, 2

Tags: cyclic quadrilateral , concyclic , geometry , equal angles

Let $ABC$ be an acute triangle and let $P,Q$ be points on $BC$ such that $\angle QAC =\angle ABC$ and $\angle PAB = \angle ACB$. We extend $AP$ to $M$ so that $ P$ is the midpoint of $AM$ and we extend $AQ$ to $N$ so that $Q$ is the midpoint of $AN$. If T is the intersection point of $BM$ and $CN$, show that quadrilateral $ABTC$ is cyclic.

2017 Brazil Team Selection Test, 2

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

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 $BDE$.

2006 Germany Team Selection Test, 3

Tags: cyclic quadrilateral , geometry , circumcircle

The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at a point $X$. The circumcircles of triangles $ABX$ and $CDX$ meet at a point $Y$ (apart from $X$). Let $O$ be the center of the circumcircle of the quadrilateral $ABCD$. Assume that the points $O$, $X$, $Y$ are all distinct. Show that $OY$ is perpendicular to $XY$.

2008 Indonesia TST, 1

Tags: cyclic quadrilateral , incenter , geometry

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 $ABD$.

2021 Oral Moscow Geometry Olympiad, 4

Tags: circumcircle , cyclic quadrilateral , geometry

On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.

1998 Iran MO (3rd Round), 2

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a convex pentagon such that \[\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.\] Let $F$ be a point on AB such that $AF:BF=AE:BC$. Show that \[\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.\]

2018 Saudi Arabia IMO TST, 3

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

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 $\angle BEC$.

2010 Sharygin Geometry Olympiad, 4

Tags: geometry , circumcircle , cyclic quadrilateral

The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N.$ The circumcircles of triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1,B_1,C_1,D_1.$ Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed in a circle centered at $N.$

2008 Ukraine Team Selection Test, 1

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

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]

2011 AMC 12/AHSME, 24

Tags: trigonometry , geometry , cyclic quadrilateral

Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, $DA=12$. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral? $ \textbf{(A)}\ \sqrt{15} \qquad\textbf{(B)}\ \sqrt{21} \qquad\textbf{(C)}\ 2\sqrt{6} \qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 2\sqrt{7} $

2018 Bosnia And Herzegovina - Regional Olympiad, 4

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

Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$

JBMO Geometry Collection, 2000

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

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]

2010 IberoAmerican, 2

Tags: circumcircle , symmetry , parallelogram , rectangle , geometry , cyclic quadrilateral , analytic geometry

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals, $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear

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)

Geometry Mathley 2011-12, 13.3

Tags: concyclic , cyclic quadrilateral , concurrency , geometry

Let $ABCD$ be a quadrilateral inscribed in circle $(O)$. Let $M,N$ be the midpoints of $AD,BC$. A line through the intersection $P$ of the two diagonals $AC,BD$ meets $AD,BC$ at $S, T$ respectively. Let $BS$ meet $AT$ at $Q$. Prove that three lines $AD,BC,PQ$ are concurrent if and only if $M, S, T,N$ are on the same circle. Đỗ Thanh Sơn

2011 India IMO Training Camp, 1

Tags: cyclic quadrilateral , trigonometry , geometry , trapezoid

Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that: $a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

2016 IMO Shortlist, G4

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

2022 Centroamerican and Caribbean Math Olympiad, 3

Tags: cyclic quadrilateral , orthocenter , circumcircle , geometry

Let $ABC$ an acutangle triangle with orthocenter $H$ and circumcenter $O$. Let $D$ the intersection of $AO$ and $BH$. Let $P$ be the point on $AB$ such that $PH=PD$. Prove that the points $B, D, O$ and $P$ lie on a circle.

2008 Postal Coaching, 5

Tags: cyclic quadrilateral , angle , geometry , equal angles , incircle

A convex quadrilateral $ABCD$ is given. There rays $BA$ and $CD$ meet in $P$, and the rays $BC$ and $AD$ meet in $Q$. Let $H$ be the projection of $D$ on $PQ$. Prove that $ABCD$ is cyclic if and only if the angle between the rays beginning at $H$ and tangent to the incircle of triangle $ADP$ is equal to the angle between the rays beginning at $H$ and tangent to the incircle of triangle $CDQ$. Also find out whether $ABCD$ is inscribable or circumscribable and justify.

2012 ELMO Shortlist, 1

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

In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$. [i]Ray Li.[/i]

1986 China Team Selection Test, 1

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

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

2015 Sharygin Geometry Olympiad, 7

Tags: tangential quadrilateral , geometry , cyclic quadrilateral

Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex M from angle $CMD$. Prove that $ABCD$ is a cyclic quadrilateral. (M. Kungozhin)

2017 Estonia Team Selection Test, 3

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

2022 Sharygin Geometry Olympiad, 5

Tags: parallel , cyclic quadrilateral , geometry

Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$. The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$, a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$.