Olympiad problems

1987 Traian Lălescu, 1.3

Let $ A'\neq A $ be the intersection of the bisector of $ \angle BAC $ with the circumcircle of the triangle $ ABC. $ Prove that $ AA'>\frac{AB+AC}{2}. $

2008 Croatia Team Selection Test, 3

Tags: circumcircle , geometry

Point $ M$ is taken on side $ BC$ of a triangle $ ABC$ such that the centroid $ T_c$ of triangle $ ABM$ lies on the circumcircle of $ \triangle ACM$ and the centroid $ T_b$ of $ \triangle ACM$ lies on the circumcircle of $ \triangle ABM$. Prove that the medians of the triangles $ ABM$ and $ ACM$ from $ M$ are of the same length.

1974 IMO Longlists, 4

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

Let $K_a,K_b,K_c$ with centres $O_a,O_b,O_c$ be the excircles of a triangle $ABC$, touching the interiors of the sides $BC,CA,AB$ at points $T_a,T_b,T_c$ respectively. Prove that the lines $O_aT_a,O_bT_b,O_cT_c$ are concurrent in a point $P$ for which $PO_a=PO_b=PO_c=2R$ holds, where $R$ denotes the circumradius of $ABC$. Also prove that the circumcentre $O$ of $ABC$ is the midpoint of the segment $PI$, where $I$ is the incentre of $ABC$.

2023 Korea Summer Program Practice Test, P3

Tags: incenter , geometry , circumcircle

$\triangle ABC$ is a triangle such that $\angle A = 60^{\circ}$. The incenter of $\triangle ABC$ is $I$. $AI$ intersects with $BC$ at $D$, $BI$ intersects with $CA$ at $E$, and $CI$ intersects with $AB$ at $F$, respectively. Also, the circumcircle of $\triangle DEF$ is $\omega$. The tangential line of $\omega$ at $E$ and $F$ intersects at $T$. Show that $\angle BTC \ge 60^{\circ}$

2011 Indonesia TST, 3

Tags: trapezoid , asymptote , perpendicular bisector , geometry , circumcircle , reflection , geometric transformation

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

2003 China Girls Math Olympiad, 7

Tags: trigonometry , circumcircle , geometry

Let the sides of a scalene triangle $ \triangle ABC$ be $ AB \equal{} c,$ $ BC \equal{} a,$ $ CA \equal{}b,$ and $ D, E , F$ be points on $ BC, CA, AB$ such that $ AD, BE, CF$ are angle bisectors of the triangle, respectively. Assume that $ DE \equal{} DF.$ Prove that (1) $ \frac{a}{b\plus{}c} \equal{} \frac{b}{c\plus{}a} \plus{} \frac{c}{a\plus{}b}$ (2) $ \angle BAC > 90^{\circ}.$

2021 SYMO, Q4

Tags: geometry , circumcircle , tangent circle

Let $ABC$ be an acute-angled triangle. The tangents to the circumcircle of triangle $ABC$ at $B$ and $C$ respectively meet at $D$. The circumcircles of triangles $ABD$ and $ACD$ meet line $BC$ at additional points $E$ and $F$ respectively. Lines $DB$ and $DC$ meet the circumcircle of triangle $DEF$ at additional points $X$ and $Y$ respectively. Let $O$ be the circumcentre of triangle $DEF$. Prove that the circumcircles of triangles $ABC$ and $OXY$ are tangent to each other.

2008 Germany Team Selection Test, 2

Tags: trapezoid , geometry , circumcircle

Let $ ABCD$ be an isosceles trapezium with $ AB \parallel{} CD$ and $ \bar{BC} \equal{} \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.

2003 National Olympiad First Round, 25

Tags: trigonometry , perpendicular bisector , circumcircle , geometry

Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$? $ \textbf{(A)}\ 56\sqrt 3 \qquad\textbf{(B)}\ 56 \sqrt 2 \qquad\textbf{(C)}\ 50 \sqrt 2 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ \text{None of the preceding} $

2012 ELMO Shortlist, 7

Tags: inversion , geometry , circumcircle , trigonometry

Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$. [i]Alex Zhu.[/i]

2003 Chile National Olympiad, 6

Tags: tangent , circumcircle , geometry

Consider a triangle $ ABC $. On the line $ AC $ take a point $ B_1 $ such that $ AB = AB_1 $ and in addition, $ B_1 $ and $ C $ are located on the same side of the line with respect to the point $ A $. The bisector of the angle $ A $ intersects the side $ BC $ at a point that we will denote as $ A_1 $. Let $ P $ and $ R $ be the circumscribed circles of the triangles $ ABC $ and $ A_1B_1C $ respectively. They intersect at points $ C $ and $ Q $. Prove that the tangent to the circle $ R $ at the point $ Q $ is parallel to the line $ AC $.

1993 IberoAmerican, 2

Tags: perpendicular bisector , combinatorics , circumcircle , rhombus , pigeonhole principle , geometry

Let $P$ and $Q$ be two distinct points in the plane. Let us denote by $m(PQ)$ the segment bisector of $PQ$. Let $S$ be a finite subset of the plane, with more than one element, that satisfies the following properties: (i) If $P$ and $Q$ are in $S$, then $m(PQ)$ intersects $S$. (ii) If $P_1Q_1, P_2Q_2, P_3Q_3$ are three diferent segments such that its endpoints are points of $S$, then, there is non point in $S$ such that it intersects the three lines $m(P_1Q_1)$, $m(P_2Q_2)$, and $m(P_3Q_3)$. Find the number of points that $S$ may contain.

2005 All-Russian Olympiad Regional Round, 11.4

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

11.4 Let $AA_1$ and $BB_1$ are altitudes of an acute non-isosceles triangle $ABC$, $A'$ is a midpoint of $BC$ and $B'$ is a midpoint of $AC$. A segement $A_1B_1$ intersects $A'B'$ at point $C'$. Prove that $CC'\perp HO$, where $H$ is a orthocenter and $O$ is a circumcenter of $ABC$. ([i]L. Emel'yanov[/i])

2019 Grand Duchy of Lithuania, 3

Tags: right angle , perpendicular , geometry , circumcircle , orthocenter

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. The perpendicular bisector of segment $CH$ intersects the sides $AC$ and $BC$ in points $X$ and $Y$ , respectively. The lines $XO$ and $YO$ intersect the side $AB$ in points $P$ and $Q$, respectively. Prove that if $XP + Y Q = AB + XY$ then $\angle OHC = 90^o$.

2002 AIME Problems, 13

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

In triangle $ ABC$ the medians $ \overline{AD}$ and $ \overline{CE}$ have lengths 18 and 27, respectively, and $ AB \equal{} 24$. Extend $ \overline{CE}$ to intersect the circumcircle of $ ABC$ at $ F$. The area of triangle $ AFB$ is $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

2014 France Team Selection Test, 2

Tags: geometry , geometric transformation , rotation , trapezoid , circumcircle , reflection , trigonometry

Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.

2015 Azerbaijan JBMO TST, 3

Tags: geometry , circumcircle

Let $ABC$ be a triangle such that $AB$ is not equal to $AC$. Let $M$ be the midpoint of $BC$ and $H$ be the orthocenter of triangle $ABC$. Let $D$ be the midpoint of $AH$ and $O$ the circumcentre of triangle $BCH$. Prove that $DAMO$ is a parallelogram.

2010 Contests, 2

Tags: parallelogram , circumcircle , geometry , trigonometry

Let $ABCD$ be a convex quadrilateral. Assume line $AB$ and $CD$ intersect at $E$, and $B$ lies between $A$ and $E$. Assume line $AD$ and $BC$ intersect at $F$, and $D$ lies between $A$ and $F$. Assume the circumcircles of $\triangle BEC$ and $\triangle CFD$ intersect at $C$ and $P$. Prove that $\angle BAP=\angle CAD$ if and only if $BD\parallel EF$.

2006 Team Selection Test For CSMO, 2

Tags: parallelogram , euler , projective geometry , perpendicular bisector , geometry , circumcircle

Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).

1999 Bosnia and Herzegovina Team Selection Test, 2

Tags: circumcircle , geometry , inequalities

Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$ in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$

1997 Iran MO (2nd round), 2

Tags: circumcircle , geometry

Let segments $KN,KL$ be tangent to circle $C$ at points $N,L$, respectively. $M$ is a point on the extension of the segment $KN$ and $P$ is the other meet point of the circle $C$ and the circumcircle of $\triangle KLM$. $Q$ is on $ML$ such that $NQ$ is perpendicular to $ML$. Prove that \[ \angle MPQ=2\angle KML. \]

2019 Romania Team Selection Test, 2

Tags: collinearity , circumcircle , geometry , incenter

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

2011 Saint Petersburg Mathematical Olympiad, 2

Tags: circumcircle , geometry

$ABC$-triangle with circumcenter $O$ and $\angle B=30$. $BO$ intersect $AC$ at $K$. $L$ - midpoint of arc $OC$ of circumcircle $KOC$, that does not contains $K$. Prove, that $A,B,L,K$ are concyclic.

2022 South Africa National Olympiad, 4

Tags: angle bisector , circumcircle , geometry

Let $ABC$ be a triangle with $AB < AC$. A point $P$ on the circumcircle of $ABC$ (on the same side of $BC$ as $A$) is chosen in such a way that $BP = CP$. Let $BP$ and the angle bisector of $\angle BAC$ intersect at $Q$, and let the line through $Q$ and parallel to $BC$ intersect $AC$ at $R$. Prove that $BR = CR$.

2000 Moldova National Olympiad, Problem 8

Tags: geometry , circumcircle

In an isosceles triangle $ABC$ with $BC=AC$ and $\angle B<60^\circ$, $I$ is the incenter and $O$ the circumcenter. The circle with center $E$ that passes through $A,O$ and $I$ intersects the circumcircle of $\triangle ABC$ again at point $D$. Prove that the lines $DE$ and $CO$ intersect on the circumcircle of $ABC$.