Olympiad problems

2006 Taiwan National Olympiad, 1

$P,Q$ are two fixed points on a circle centered at $O$, and $M$ is an interior point of the circle that differs from $O$. $M,P,Q,O$ are concyclic. Prove that the bisector of $\angle PMQ$ is perpendicular to line $OM$.

Ukrainian TYM Qualifying - geometry, XI.15

Tags: geometry , incenter , inequalities , circumcircle

Let $I$ be the point of intersection of the angle bisectors of the $\vartriangle ABC$, $W_1,W_2,W_3$ be point of intersection of lines $AI, BI, CI$ with the circle circumscribed around the triangle, $r$ and $R$ be the radii of inscribed and circumscribed circles respectively. Prove the inequality $$IW_1+ IW_2 + IW_3\ge 2R + \sqrt{2Rr.}$$

1995 Mexico National Olympiad, 2

Tags: geometry , circumcircle , combinatorics

Consider 6 points on a plane such that 8 of the distances between them are equal to 1. Prove that there are at least 3 points that form an equilateral triangle.

2013 ELMO Shortlist, 12

Tags: circumcircle , ratio , geometric transformation , reflection , geometry

Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$. [i]Proposed by David Stoner[/i]

2016 Hong Kong TST, 3

Tags: circumcircle , geometry

Let $ABC$ be a triangle such that $AB \neq AC$. The incircle with centre $I$ touches $BC$ at $D$. Line $AI$ intersects the circumcircle $\Gamma$ of $ABC$ at $M$, and $DM$ again meets $\Gamma$ at $P$. Find $\angle API$

2022 Turkey MO (2nd round), 1

Tags: circumcircle , geometry

In triangle $ABC$, $M$ is the midpoint of side $BC$, the bisector of angle $BAC$ intersects $BC$ and $(ABC)$ at $K$ and $L$, respectively. If the circle with diameter $[BC]$ is tangent to the external angle bisector of angle $BAC$, prove that this circle is tangent to $(KLM)$ as well.

2013 Sharygin Geometry Olympiad, 9

Tags: geometry , geometric transformation , incenter , circumcircle , asymptote , analytic geometry , reflection

Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$.

2014 All-Russian Olympiad, 4

Tags: incenter , circumcircle , trigonometry , angle bisector , perpendicular bisector , geometry

Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. [i]M. Kungodjin[/i]

2014 Iran Team Selection Test, 2

Tags: geometry , incenter , circumcircle , geometric transformation

Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.

2005 Mediterranean Mathematics Olympiad, 2

Tags: circumcircle , geometry

Let $k$ and $k'$ be concentric circles with center $O$ and radius $R$ and $R'$ where $R<R'$ holds. A line passing through $O$ intersects $k$ at $A$ and $k'$ at $B$ where $O$ is between $A$ and $B$. Another line passing through $O$ and distict from $AB$ intersects $k$ at $E$ and $k'$ at $F$ where $E$ is between $O$ and $F$. Prove that the circumcircles of the triangles $OAE$ and $OBF$, the circle with diameter $EF$ and the circle with diameter $AB$ are concurrent.

2014 India IMO Training Camp, 3

Tags: circumcircle , geometric transformation , trigonometry , function , angle bisector , geometry

In a triangle $ABC$, points $X$ and $Y$ are on $BC$ and $CA$ respectively such that $CX=CY$,$AX$ is not perpendicular to $BC$ and $BY$ is not perpendicular to $CA$.Let $\Gamma$ be the circle with $C$ as centre and $CX$ as its radius.Find the angles of triangle $ABC$ given that the orthocentres of triangles $AXB$ and $AYB$ lie on $\Gamma$.

2024 Oral Moscow Geometry Olympiad, 3

Tags: geometry , angle bisector , circumcircle

The hypotenuse $AB$ of a right-angled triangle $ABC$ touches the corresponding excircle $\omega$ at point $T$. Point $S$ is symmetrical $T$ relative to the bisector of angle $C$, $CH$ is the height of the triangle. Prove that the circumcircle of triangle $CSH$ touches the circle $\omega$.

2018 Peru IMO TST, 7

Tags: geometry , circumcircle

Let $ABC$ be, with $AC>AB$, an acute-angled triangle with circumcircle $\Gamma$ and $M$ the midpoint of side $BC$. Let $N$ be a point in the interior of $\bigtriangleup ABC$. Let $D$ and $E$ be the feet of the perpendiculars from $N$ to $AB$ and $AC$, respectively. Suppose that $DE\perp AM$. The circumcircle of $\bigtriangleup ADE$ meets $\Gamma$ at $L$ ($L\neq A$), lines $AL$ and $DE$ intersects at $K$ and line $AN$ meets $\Gamma$ at $F$ ($F\neq A$). Prove that if $N$ is the midpoint of the segment $AF$ then $KA=KF$.

2000 Vietnam Team Selection Test, 1

Tags: geometry , circumcircle

Two circles $C_{1}$ and $C_{2}$ intersect at points $P$ and $Q$. Their common tangent, closer to $P$ than to $Q$, touches $C_{1}$ at $A$ and $C_{2}$ at $B$. The tangents to $C_{1}$ and $C_{2}$ at $P$ meet the other circle at points $E \not = P$ and $F \not = P$ , respectively. Let $H$ and $K$ be the points on the rays $AF$ and $BE$ respectively such that $AH = AP$ and $BK = BP$ . Prove that $A,H,Q,K,B$ lie on a circle.

1989 Austrian-Polish Competition, 8

Tags: geometry , circumcircle , inradius , incenter , euler , inequalities , area of a triangle

$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$, but that for any other triangle it is not.

2004 IberoAmerican, 2

Tags: geometry , circumcircle , induction , ratio , geometric transformation , reflection , power of a point

In the plane are given a circle with center $ O$ and radius $ r$ and a point $ A$ outside the circle. For any point $ M$ on the circle, let $ N$ be the diametrically opposite point. Find the locus of the circumcenter of triangle $ AMN$ when $ M$ describes the circle.

Mathley 2014-15, 6

Tags: geometry , circumcircle , bicentric quadrilateral , tangent circle

A quadrilateral is called bicentric if it has both an incircle and a circumcircle. $ABCD$ is a bicentric quadrilateral with $(O)$ being its circumcircle. Let $E, F$ be the intersections of $AB$ and $CD, AD$ and $BC$ respectively. Prove that there is a circle with center $O$ tangent to all of the circumcircles of the four triangles $EAD, EBC, FAB, FCD$. Nguyen Van Linh, a student of the Vietnamese College, Ha Noi

1996 All-Russian Olympiad, 6

Tags: circumcircle , incenter , geometry

In the isosceles triangle $ABC$ ($AC = BC$) point $O$ is the circumcenter, $I$ the incenter, and $D$ lies on $BC$ so that lines $OD$ and $BI$ are perpendicular. Prove that $ID$ and $AC$ are parallel. [i]M. Sonkin[/i]

2010 India IMO Training Camp, 1

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

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

2016 Sharygin Geometry Olympiad, P16

Tags: geometry , circumcircle , euler line

Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.

2000 All-Russian Olympiad, 7

Tags: geometry , circumcircle , parallelogram , geometric transformation , homothety

Two circles are internally tangent at $N$. The chords $BA$ and $BC$ of the larger circle are tangent to the smaller circle at $K$ and $M$ respectively. $Q$ and $P$ are midpoint of arcs $AB$ and $BC$ respectively. Circumcircles of triangles $BQK$ and $BPM$ are intersect at $L$. Show that $BPLQ$ is a parallelogram.

2003 Serbia Team Selection Test, 2

Tags: geometry , circumcircle

Let M and N be the distinct points in the plane of the triangle ABC such that AM : BM : CM = AN : BN : CN. Prove that the line MN contains the circumcenter of △ABC.

2012 Harvard-MIT Mathematics Tournament, 8

Tags: geometry , hmmt , circumcircle , incircle , hexagon

Hexagon $ABCDEF$ has a circumscribed circle and an inscribed circle. If $AB = 9$, $BC = 6$, $CD = 2$, and $EF = 4$. Find $\{DE, FA\}$.

1997 IberoAmerican, 2

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

In an acute triangle $\triangle{ABC}$, let $AE$ and $BF$ be highs of it, and $H$ its orthocenter. The symmetric line of $AE$ with respect to the angle bisector of $\sphericalangle{A}$ and the symmetric line of $BF$ with respect to the angle bisector of $\sphericalangle{B}$ intersect each other on the point $O$. The lines $AE$ and $AO$ intersect again the circuncircle to $\triangle{ABC}$ on the points $M$ and $N$ respectively. Let $P$ be the intersection of $BC$ with $HN$; $R$ the intersection of $BC$ with $OM$; and $S$ the intersection of $HR$ with $OP$. Show that $AHSO$ is a paralelogram.

2016 Croatia Team Selection Test, Problem 3

Tags: circumcircle , incircle , geometry

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.