Olympiad problems

2007 IMO Shortlist, 1

In triangle $ ABC$ the bisector of angle $ BCA$ intersects the circumcircle again at $ R$, the perpendicular bisector of $ BC$ at $ P$, and the perpendicular bisector of $ AC$ at $ Q$. The midpoint of $ BC$ is $ K$ and the midpoint of $ AC$ is $ L$. Prove that the triangles $ RPK$ and $ RQL$ have the same area. [i]Author: Marek Pechal, Czech Republic[/i]

2010 China Second Round Olympiad, 1

Tags: geometry , circumcircle

Given an acute triangle whose circumcenter is $O$.let $K$ be a point on $BC$,different from its midpoint.$D$ is on the extension of segment $AK,BD$ and $AC$,$CD$and$AB$intersect at $N,M$ respectively.prove that $A,B,D,C$ are concyclic.

2013 Stanford Mathematics Tournament, 24

Tags: circumcircle , incenter , geometry

Compute the square of the distance between the incenter (center of the inscribed circle) and circumcenter (center of the circumscribed circle) of a 30-60-90 right triangle with hypotenuse of length 2.

2009 Greece Team Selection Test, 2

Tags: perpendicular bisector , circumcircle , geometry

Given is a triangle $ABC$ with barycenter $G$ and circumcenter $O$.The perpendicular bisectors of $GA,GB,GC$ intersect at $A_1,B_1,C_1$.Show that $O$ is the barycenter of $\triangle{A_1B_1C_1}$.

Cono Sur Shortlist - geometry, 2012.G2

Tags: geometry , circumcircle , power of a point , perpendicular bisector

Let $ABC$ be a triangle, and $M$ and $N$ variable points on $AB$ and $AC$ respectively, such that both $M$ and $N$ do not lie on the vertices, and also, $AM \times MB = AN \times NC$. Prove that the perpendicular bisector of $MN$ passes through a fixed point.

2014 Iran MO (3rd Round), 5

Tags: circumcircle , geometry

$X$ and $Y$ are two points lying on or on the extensions of side $BC$ of $\triangle{ABC}$ such that $\widehat{XAY} = 90$. Let $H$ be the orthocenter of $\triangle{ABC}$. Take $X'$ and $Y'$ as the intersection points of $(BH,AX)$ and $(CH,AY)$ respectively. Prove that circumcircle of $\triangle{CYY'}$,circumcircle of $\triangle{BXX'}$ and $X'Y'$ are concurrent.

1993 APMO, 1

Tags: geometry , circumcircle

Let $ABCD$ be a quadrilateral such that all sides have equal length and $\angle{ABC} =60^o$. Let $l$ be a line passing through $D$ and not intersecting the quadrilateral (except at $D$). Let $E$ and $F$ be the points of intersection of $l$ with $AB$ and $BC$ respectively. Let $M$ be the point of intersection of $CE$ and $AF$. Prove that $CA^2 = CM \times CE$.

1978 IMO Longlists, 41

Tags: circumcircle , triangle , geometry , inradius , incenter

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

2012 All-Russian Olympiad, 2

Tags: perpendicular bisector , circumcircle , reflection , geometry , geometric transformation , inradius

The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.

2021 IMO Shortlist, G5

Tags: circumcircle , projective geometry , geometry

Let $ABCD$ be a cyclic quadrilateral whose sides have pairwise different lengths. Let $O$ be the circumcenter of $ABCD$. The internal angle bisectors of $\angle ABC$ and $\angle ADC$ meet $AC$ at $B_1$ and $D_1$, respectively. Let $O_B$ be the center of the circle which passes through $B$ and is tangent to $\overline{AC}$ at $D_1$. Similarly, let $O_D$ be the center of the circle which passes through $D$ and is tangent to $\overline{AC}$ at $B_1$. Assume that $\overline{BD_1} \parallel \overline{DB_1}$. Prove that $O$ lies on the line $\overline{O_BO_D}$.

2013 ELMO Shortlist, 2

Tags: reflection , trapezoid , geometric transformation , circumcircle , incenter , geometry

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$, and let $D$,$E$,$F$ be the points where its incircle meets $BC$, $AC$, $AB$ respectively. Let the circumcircles of $\triangle AEF$, $\triangle BFD$, and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent. [i]Proposed by Michael Kural[/i]

1997 Iran MO (3rd Round), 2

Tags: trigonometry , circumcircle , geometry , inequalities

Show that for any arbitrary triangle $ABC$, we have \[\sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \leq \frac{abc}{(a+b)(b+c)(c+a)}.\]

2010 Contests, 2

Tags: lalala , geometry , analytic geometry , rotation , perpendicular bisector , circumcircle

$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.

2024 Argentina Cono Sur TST, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle. The point $B'$ of the line $CA$ is such that $A$, $C$ and $B'$ are in that order on the line and $B'C=AB$; the point $C'$ of the line $AB$ is such that $A$, $B$ and $C'$ are in that order on the line and $C'B=AC$. Prove that the circumcenter of triangle $AB'C'$ belongs to the circumcircle of triangle $ABC$.

2010 Sharygin Geometry Olympiad, 5

Tags: right triangle , equal segments , altitude , circumcircle , geometry

Let $BH$ be an altitude of a right-angled triangle $ABC$ ($\angle B = 90^o$). The incircle of triangle $ABH$ touches $AB,AH$ in points $H_1, B_1$, the incircle of triangle $CBH$ touches $CB,CH$ in points $H_2, B_2$, point $O$ is the circumcenter of triangle $H_1BH_2$. Prove that $OB_1 = OB_2$.

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]

1997 USAMO, 2

Tags: trigonometry , radical axis , circumcircle , geometry

Let $ABC$ be a triangle. Take points $D$, $E$, $F$ on the perpendicular bisectors of $BC$, $CA$, $AB$ respectively. Show that the lines through $A$, $B$, $C$ perpendicular to $EF$, $FD$, $DE$ respectively are concurrent.

2002 ITAMO, 3

Tags: circumcircle , geometry , perpendicular bisector , trigonometry

Let $A$ and $B$ are two points on a plane, and let $M$ be the midpoint of $AB$. Let $r$ be a line and let $R$ and $S$ be the projections of $A$ and $B$ onto $r$. Assuming that $A$, $M$, and $R$ are not collinear, prove that the circumcircle of triangle $AMR$ has the same radius as the circumcircle of $BSM$.

2014 Oral Moscow Geometry Olympiad, 5

Tags: geometry , area , equilateral , equilateral triangle , circumcircle , area of a triangle

Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

2010 Sharygin Geometry Olympiad, 7

Tags: circumcircle , geometry

The line passing through the vertex $B$ of a triangle $ABC$ and perpendicular to its median $BM$ intersects the altitudes dropped from $A$ and $C$ (or their extensions) in points $K$ and $N.$ Points $O_1$ and $O_2$ are the circumcenters of the triangles $ABK$ and $CBN$ respectively. Prove that $O_1M=O_2M.$

JBMO Geometry Collection, 2019

Tags: circumcircle , geometry , perpendicular bisector

Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.

2024 Sharygin Geometry Olympiad, 8.6

Tags: geometry , reflection , circumcircle

A circle $\omega$ touched lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle meets $a$ and $b$ at $X$ and $Y$ respectively. Points $X'$ and $Y'$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Find the locus of projections of the center of the circle to the lines $X'Y'$.

2005 Moldova Team Selection Test, 1

Tags: law of sines , trig identities , inradius , circumcircle , geometry , trigonometry , inequalities

Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that $\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$, where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.

2009 Saint Petersburg Mathematical Olympiad, 2

Tags: circumcircle , geometry

$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.

1998 All-Russian Olympiad, 6

Tags: parallelogram , circumcircle , trapezoid , angle bisector , geometry

In triangle $ABC$ with $AB>BC$, $BM$ is a median and $BL$ is an angle bisector. The line through $M$ and parallel to $AB$ intersects $BL$ at point $D$, and the line through $L$ and parallel to $BC$ intersects $BM$ at point $E$. Prove that $ED$ is perpendicular to $BL$.