Olympiad problems

2004 Oral Moscow Geometry Olympiad, 5

The diagonals of the inscribed quadrilateral $ABCD$ meet at the point $M$, $\angle AMB = 60^o$. Equilateral triangles $ADK$ and $BCL$ are built outward on sides $AD$ and $BC$. Line $KL$ meets the circle circumscribed ariound $ABCD$ at points $P$ and $Q$. Prove that $PK = LQ$.

2016 Regional Olympiad of Mexico Southeast, 2

Tags: geometry , trapezoid , circumcircle , cyclic quadrilateral

Let $ABCD$ a trapezium with $AB$ parallel to $CD, \Omega$ the circumcircle of $ABCD$ and $A_1,B_1$ points on segments $AC$ and $BC$ respectively, such that $DA_1B_1C$ is a cyclic cuadrilateral. Let $A_2$ and $B_2$ the symmetric points of $A_1$ and $B_1$ with respect of the midpoint of $AC$ and $BC$, respectively. Prove that points $A, B, A_2, B_2$ are concyclic.

2011 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , geometry , incenter , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$.

2003 Turkey Junior National Olympiad, 1

Tags: geometry , trigonometry , quadratic , cyclic quadrilateral , trig identities , law of cosines

Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$, $|BD|=6$, and $|AD|\cdot|CE|=|DC|\cdot|AE|$, find the area of the quadrilateral $ABCD$.

2022 Macedonian Mathematical Olympiad, Problem 2

Tags: geometry , cyclic quadrilateral , circumcircle , 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]

2008 Sharygin Geometry Olympiad, 9

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

(A.Zaslavsky, 9--10) The reflections of diagonal $ BD$ of a quadrilateral $ ABCD$ in the bisectors of angles $ B$ and $ D$ pass through the midpoint of diagonal $ AC$. Prove that the reflections of diagonal $ AC$ in the bisectors of angles $ A$ and $ C$ pass through the midpoint of diagonal $ BD$ (There was an error in published condition of this problem).

2015 Tournament of Towns, 4

Tags: geometry , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, $K$ and $N$ be the midpoints of the diagonals and $P$ and $Q$ be points of intersection of the extensions of the opposite sides. Prove that $\angle PKQ + \angle PNQ = 180$. [i]($7$ points)[/i] .

2006 Turkey MO (2nd round), 1

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

Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.

2020 Tournament Of Towns, 5

Tags: geometry , cyclic quadrilateral , circles , concurrency

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

2017 Singapore Senior Math Olympiad, 2

Tags: geometry , right angle , perpendicular , cyclic quadrilateral , cyclic

In the cyclic quadrilateral $ABCD$, the sides $AB, DC$ meet at $Q$, the sides $AD,BC$ meet at $P, M$ is the midpoint of $BD$, If $\angle APQ=90^o$, prove that $PM$ is perpendicular to $AB$.

2008 Indonesia TST, 3

Tags: geometry , cyclic quadrilateral , angle bisector , tangent circle

Let $\Gamma_1$ and $\Gamma_2$ be two circles that tangents each other at point $N$, with $\Gamma_2$ located inside $\Gamma_1$. Let $A, B, C$ be distinct points on $\Gamma_1$ such that $AB$ and $AC$ tangents $\Gamma_2$ at $D$ and $E$, respectively. Line $ND$ cuts $\Gamma_1$ again at $K$, and line $CK$ intersects line $DE$ at $I$. (i) Prove that $CK$ is the angle bisector of $\angle ACB$. (ii) Prove that $IECN$ and $IBDN$ are cyclic quadrilaterals.

2009 Ukraine National Mathematical Olympiad, 3

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

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

2020 Regional Olympiad of Mexico Southeast, 2

Tags: geometry , incenter , circumcircle , cyclic quadrilateral

Let $ABC$ a triangle with $AB<AC$ and let $I$ it´s incenter. Let $\Gamma$ the circumcircle of $\triangle BIC$. $AI$ intersect $\Gamma$ again in $P$. Let $Q$ a point in side $AC$ such that $AB=AQ$ and let $R$ a point in $AB$ with $B$ between $A$ and $R$ such that $AR=AC$. Prove that $IQPR$ is cyclic.

2012 Estonia Team Selection Test, 3

Tags: diagonal , cyclic quadrilateral , acute , geometry , circumcircle

In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.

2007 Romania Team Selection Test, 4

Tags: euler , inequalities , circumcircle , geometric transformation , homothety , cyclic quadrilateral , geometry

The points $M, N, P$ are chosen on the sides $BC, CA, AB$ of a triangle $\Delta ABC$, such that the triangle $\Delta MNP$ is acute-angled. We denote with $x$ the length of the shortest altitude of the triangle $\Delta ABC$, and with $X$ the length of the longest altitudes of the triangle $\Delta MNP$. Prove that $x \leq 2X$.

2021 Chile National Olympiad, 4

Tags: geometry , concyclic , cyclic quadrilateral

Consider quadrilateral $ABCD$ with $|DC| > |AD|$. Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$. Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $ \angle ABC$. Let us call $X = L_1 \cap L_2$. Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$ [img]https://cdn.artofproblemsolving.com/attachments/f/6/3ebfce8a7fd2a0ece9f09065608141006893d2.png[/img]

2012 ELMO Problems, 1

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

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]

2007 Sharygin Geometry Olympiad, 19

Tags: geometry , minimum , geometric inequality , angle , cyclic quadrilateral , area

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

2003 IMO Shortlist, 1

Tags: geometry , angle bisector , cyclic quadrilateral , quadrilateral , concurrency

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2005 Sharygin Geometry Olympiad, 11.3

Tags: diagonal , ratio , distance , cyclic quadrilateral , geometry

Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.

Russian TST 2016, P1

Tags: geometry , cyclic quadrilateral

In the cyclic quadrilateral $ABCD$, the diagonal $BD$ is divided in half by the diagonal $AC$. The points $E, F, G$ and $H{}$ are the midpoints of the sides $AB, BC, CD{}$ and $DA$ respectively. Let $P = AD \cap BC$ and $Q = AB \cap CD{}$. The bisectors of the angles $APC$ and $AQC$ intersect the segments $EG$ and $FH$ at the points $X{}$ and $Y{}$ respectively. Prove that $XY \parallel BD$.

2010 Sharygin Geometry Olympiad, 4

Tags: circumcircle , cyclic quadrilateral , geometry

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 $N.$

1998 India National Olympiad, 4

Tags: trigonometry , rectangle , cyclic quadrilateral , geometric inequality , geometry

Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.

2022 Malaysia IMONST 2, 1

Tags: geometry , cyclic quadrilateral

Given a circle and a quadrilateral $ABCD$ whose vertices all lie on the circle. Let $R$ be the midpoint of arc $AB$. The line $RC$ meets line $AB$ at point $S$, and the line $RD$ meets line $AB$ at point $T$. Prove that $CDTS$ is a cyclic quadrilateral.

2013 ELMO Shortlist, 9

Tags: rectangle , symmetry , circumcircle , cyclic quadrilateral , projective geometry , geometry

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]