Olympiad problems

2019 Iranian Geometry Olympiad, 3

Circles $\omega_1$ and $\omega_2$ have centres $O_1$ and $O_2$, respectively. These two circles intersect at points $X$ and $Y$. $AB$ is common tangent line of these two circles such that $A$ lies on $\omega_1$ and $B$ lies on $\omega_2$. Let tangents to $\omega_1$ and $\omega_2$ at $X$ intersect $O_1O_2$ at points $K$ and $L$, respectively. Suppose that line $BL$ intersects $\omega_2$ for the second time at $M$ and line $AK$ intersects $\omega_1$ for the second time at $N$. Prove that lines $AM, BN$ and $O_1O_2$ concur. [i]Proposed by Dominik Burek - Poland[/i]

1972 IMO Shortlist, 10

Tags: cyclic quadrilateral , trapezoid , dissection , geometry

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

1989 China Team Selection Test, 4

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

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 $P_2 Q_2 R_2$.

2017 NIMO Problems, 3

Tags: cyclic quadrilateral , geometry

Let $ABCD$ be a cyclic quadrilateral with circumradius $100\sqrt{3}$ and $AC=300$. If $\angle DBC = 15^{\circ}$, then find $AD^2$. [i]Proposed by Anand Iyer[/i]

2012 Kazakhstan National Olympiad, 2

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

Let $ABCD$ be an inscribed quadrilateral, in which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $

2021 Dutch BxMO TST, 1

Tags: geometry , parallel , cyclic quadrilateral

Given is a cyclic quadrilateral $ABCD$ with $|AB| = |BC|$. Point $E$ is on the arc $CD$ where $A$ and $B$ are not on. Let $P$ be the intersection point of $BE$ and $CD$ , let $Q$ be the intersection point of $AE$ and $BD$ . Prove that $PQ \parallel AC$.

2022 Sharygin Geometry Olympiad, 8.4

Tags: geometry , area , equal area , cyclic quadrilateral

Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.

2024 Baltic Way, 11

Tags: cyclic quadrilateral , geometry , circumcircle

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O$ and with $AC$ perpendicular to $BD$. Points $X$ and $Y$ lie on the circumcircle of the triangle $BOD$ such that $\angle AXO=\angle CYO=90^{\circ}$. Let $M$ be the midpoint of $AC$. Prove that $BD$ is tangent to the circumcircle of the triangle $MXY$.

II Soros Olympiad 1995 - 96 (Russia), 10.9

Tags: cyclic quadrilateral , newton line , geometry

The opposite sides of a quadrilateral inscribed in a circle intersect at points $K$ and $L$. Let $F$ be the midpoint of $KL$, $E$ and $G$ be the midpoints of the diagonals of the given quadrilateral. It is known that $FE = a$, $FG = b$. Calculate $KL$ in terms of $a$ and $b.$ (It is known that the points $F$, $E$ and $G$ lie on the same straight line. This is true for any quadrilateral, not necessarily inscribed. The indicated straight line is sometimes called the Newton−Gauss line. This fact can be used without proof in proving the problem, as it is known).

2015 CCA Math Bonanza, L3.4

Tags: cyclic quadrilateral , geometry , inequalities

Compute the greatest constant $K$ such that for all positive real numbers $a,b,c,d$ measuring the sides of a cyclic quadrilateral, we have \[ \left(\frac{1}{a+b+c-d}+\frac{1}{a+b-c+d}+\frac{1}{a-b+c+d}+\frac{1}{-a+b+c+d}\right)(a+b+c+d)\geq K. \] [i]2015 CCA Math Bonanza Lightning Round #3.4[/i]

JBMO Geometry Collection, 2007

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

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2024 Bangladesh Mathematical Olympiad, P9

Tags: cyclic quadrilateral , inversion , spiral similarity , arc midpoint , power of a point , geometry

Let $ABC$ be a triangle and $M$ be the midpoint of side $BC$. The perpendicular bisector of $BC$ intersects the circumcircle of $\triangle ABC$ at points $K$ and $L$ ($K$ and $A$ lie on the opposite sides of $BC$). A circle passing through $L$ and $M$ intersects $AK$ at points $P$ and $Q$ ($P$ lies on the line segment $AQ$). $LQ$ intersects the circumcircle of $\triangle KMQ$ again at $R$. Prove that $BPCR$ is a cyclic quadrilateral.

2023 Switzerland - Final Round, 7

Tags: cyclic quadrilateral , arc midpoint , 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.

2013 AMC 12/AHSME, 19

Tags: cyclic quadrilateral , similar triangles , geometry , analytic geometry , ratio

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? ${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $

2016 Moldova Team Selection Test, 11

Tags: cyclic quadrilateral , geometry

Let $ABCD$ be a cyclic quadrilateral. Circle with diameter $AB$ intersects $CA$, $CB$, $DA$, and $DB$ in $E$, $F$, $G$, and $H$, respectively (all different from $A$ and $B$). The lines $EF$ and $GH$ intersect in $I$. Prove that the bisector of $\angle GIF$ and the line $CD$ are perpendicular.

2020 Dutch BxMO TST, 2

Tags: cyclic quadrilateral , geometry , concyclic

In an acute-angled triangle $ABC, D$ is the foot of the altitude from $A$. Let $D_1$ and $D_2$ be the symmetric points of $D$ wrt $AB$ and $AC$, respectively. Let $E_1$ be the intersection of $BC$ and the line through $D_1$ parallel to $AB$ . Let $E_2$ be the intersection of$ BC$ and the line through $D_2$ parallel to $AC$. Prove that $D_1, D_2, E_1$ and $E_2$ on one circle whose center lies on the circumscribed circle of $\vartriangle ABC$.

2022 Malaysia IMONST 2, 1

Tags: cyclic quadrilateral , geometry

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.

2009 Moldova Team Selection Test, 3

Tags: cyclic quadrilateral , circumcircle , search , geometry

[color=darkred]A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.[/color]

2011 India IMO Training Camp, 1

Tags: trapezoid , trigonometry , cyclic quadrilateral , geometry

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. $b) a\cdot AP=b\cdot BP=c\cdot PC.$

2017 Estonia Team Selection Test, 3

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

2017 Sharygin Geometry Olympiad, P22

Tags: cyclic quadrilateral , projection , distance , geometry

Let $P$ be an arbitrary point on the diagonal $AC$ of cyclic quadrilateral $ABCD$, and $PK, PL, PM, PN, PO$ be the perpendiculars from $P$ to $AB, BC, CD, DA, BD$ respectively. Prove that the distance from $P$ to $KN$ is equal to the distance from $O$ to $ML$.

2013 Estonia Team Selection Test, 4

Tags: right triangle , cyclic quadrilateral , concyclic , square , geometry

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

2009 Puerto Rico Team Selection Test, 5

Tags: cyclic quadrilateral , geometry

Let $ ABCD$ be a quadrilateral inscribed in a circle. The diagonal $ BD$ bisects $ AC$. If $ AB = 10$, $ AD = 12$ and $ DC = 11$, find $ BC$.

1999 Turkey Team Selection Test, 2

Tags: reflection , circumcircle , geometric transformation , cyclic quadrilateral , ratio , geometry , angle bisector

Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.

1972 Bulgaria National Olympiad, Problem 5

Tags: perpendicular , cyclic quadrilateral , geometric inequalities , geometry , inequalities

In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral. (a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed. (b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold? [i]H. Lesov[/i]