Olympiad problems

2014 NIMO Problems, 5

Tags: circumcircle , ratio , trigonometry , complex number , pythagorean theorem , geometry

Triangle $ABC$ has sidelengths $AB = 14, BC = 15,$ and $CA = 13$. We draw a circle with diameter $AB$ such that it passes $BC$ again at $D$ and passes $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\tfrac{m}{n}$ where $m, n$ are coprime positive integers, determine $100m+n$. [i]Proposed by Lewis Chen[/i]

2024 All-Russian Olympiad, 6

Tags: circumcircle , geometry

The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$. [i]Proposed by A. Tereshin[/i]

2016 IberoAmerican, 3

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$. Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$, and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$. Prove that $L, J, A, Q,$ and $K$ all lie on a circle.

2004 AMC 10, 24

Tags: ratio , circumcircle , angle bisector , power of a point , geometry

In $ \triangle ABC$ we have $ AB \equal{} 7$, $ AC \equal{} 8$, and $ BC \equal{} 9$. Point $ D$ is on the circumscribed circle of the triangle so that $ \overline{AD}$ bisects $ \angle BAC$. What is the value of $ AD/CD$? $ \textbf{(A)}\ \frac{9}{8}\qquad \textbf{(B)}\ \frac{5}{3}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{17}{7}\qquad \textbf{(E)}\ \frac{5}{2}$

2018 Estonia Team Selection Test, 7

Tags: geometry , altitude , equal segments , circumcircle , concurrency

Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.

2012 ELMO Shortlist, 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]

1981 Vietnam National Olympiad, 1

Tags: trigonometry , perimeter , circumcircle , inradius , geometry

Prove that a triangle $ABC$ is right-angled if and only if \[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]

2019 China Team Selection Test, 5

Tags: geometry , incenter , circumcircle

In $\Delta ABC$, $AD \perp BC$ at $D$. $E,F$ lie on line $AB$, such that $BD=BE=BF$. Let $I,J$ be the incenter and $A$-excenter. Prove that there exist two points $P,Q$ on the circumcircle of $\Delta ABC$ , such that $PB=QC$, and $\Delta PEI \sim \Delta QFJ$ .

2024 Macedonian Balkan MO TST, Problem 2

Tags: geometry , circumcircle

Let $D$ and $E$ be points on the sides $BC$ and $AC$ of the triangle $\triangle ABC$, respectively. The circumcircle of $\triangle ADC$ meets the circumcircle of $\triangle BCE$ for the second time at $F$. The line $FE$ meets the line $AD$ at $G$, while the line $FD$ meets the line $BE$ at $H$. Prove that the lines $CF$, $AH$ and $BG$ pass through the same point. [i]Authored by Petar Filipovski[/i]

2018 Yasinsky Geometry Olympiad, 6

Tags: circumcircle , incenter , perpendicularity , perpendicular , isosceles , isosceles triangle , geometry

Given a triangle $ABC$, in which $AB = BC$. Point $O$ is the center of the circumcircle, point $I$ is the center of the incircle. Point $D$ lies on the side $BC$, such that the lines $DI$ and $AB$ parallel. Prove that the lines $DO$ and $CI$ are perpendicular. (Vyacheslav Yasinsky)

2006 All-Russian Olympiad, 4

Tags: circumcircle , geometric transformation , homothety , ratio , power of a point , radical axis , geometry

Given a triangle $ ABC$. The angle bisectors of the angles $ ABC$ and $ BCA$ intersect the sides $ CA$ and $ AB$ at the points $ B_1$ and $ C_1$, and intersect each other at the point $ I$. The line $ B_1C_1$ intersects the circumcircle of triangle $ ABC$ at the points $ M$ and $ N$. Prove that the circumradius of triangle $ MIN$ is twice as long as the circumradius of triangle $ ABC$.

2000 National High School Mathematics League, 1

Tags: geometry , circumcircle

In acute triangle $ABC$, $D,E$ are two points on side $BC$, satisfying that $\angle BAE=\angle CAF$. $FM\perp AB,EN\perp AC$ ($M,N$ are foot points). $AE$ intersects the circumcircle of $\triangle ABC$ at $D$. Prove that the area of $\triangle ABC$ and quadrilateral $AMDN$ are equal.

2011 AMC 12/AHSME, 20

Tags: geometry , circumcircle , trigonometry

Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $ X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA+XB+XC$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4} $

2009 IMO Shortlist, 6

Tags: geometry , circumcircle , trigonometry , symmetry

Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent. [i]Proposed by Eugene Bilopitov, Ukraine[/i]

2005 China Team Selection Test, 3

Tags: rhombus , circumcircle , trapezoid , perpendicular bisector , geometry

Find the least positive integer $n$ ($n\geq 3$), such that among any $n$ points (no three are collinear) in the plane, there exist three points which are the vertices of a non-isoscele triangle.

2014 Contests, 3

Tags: circumcircle , parallelogram , ellipse , perpendicular bisector , geometry , conic

Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.

Durer Math Competition CD Finals - geometry, 2018.C+2

Tags: geometry , incenter , circumcircle , equal segments

Given an $ABC$ triangle. Let $D$ be an extension of section $AB$ beyond $A$ such that that $AD = BC$ and $E$ is the extension of the section $BC$ beyond $B$ such that $BE = AC$. Prove that the circumcircle of triangle $DEB$ passes through the center of the inscribed circle of triangle $ABC$.

2011 IMO Shortlist, 4

Tags: circumcircle , symmetry , homothety , radical axis , marvio , geometry

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

2022 Kyiv City MO Round 1, Problem 3

Tags: geometry , orthocenter , circumcenter , circumcircle

Let $H$ and $O$ be the orthocenter and the circumcenter of the triangle $ABC$. Line $OH$ intersects the sides $AB, AC$ at points $X, Y$ correspondingly, so that $H$ belongs to the segment $OX$. It turned out that $XH = HO = OY$. Find $\angle BAC$. [i](Proposed by Oleksii Masalitin)[/i]

2015 Tuymaada Olympiad, 7

Tags: geometry , circumcircle

In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$. Find possible values of $\angle CED$ [i]D. Shiryaev [/i]

Kyiv City MO Seniors Round2 2010+ geometry, 2010.10.4

Tags: fixed , geometry , fixed point , length , circumcircle , angle

The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that: a) the segment $MN$ has a constant length, b) all circles circumscribed around triangle $DMN$ have a common point

2019 Mexico National Olympiad, 6

Tags: geometry , circumcircle

Let $ABC$ be a triangle such that $\angle BAC = 45^{\circ}$. Let $H,O$ be the orthocenter and circumcenter of $ABC$, respectively. Let $\omega$ be the circumcircle of $ABC$ and $P$ the point on $\omega$ such that the circumcircle of $PBH$ is tangent to $BC$. Let $X$ and $Y$ be the circumcenters of $PHB$ and $PHC$ respectively. Let $O_1,O_2$ be the circumcenters of $PXO$ and $PYO$ respectively. Prove that $O_1$ and $O_2$ lie on $AB$ and $AC$, respectively.

2010 Estonia Team Selection Test, 4

Tags: geometry , circumcircle , angle bisector , perpendicular , tangent

In an acute triangle $ABC$ the angle $C$ is greater than the angle $A$. Let $AE$ be a diameter of the circumcircle of the triangle. Let the intersection point of the ray $AC$ and the tangent of the circumcircle through the vertex $B$ be $K$. The perpendicular to $AE$ through $K$ intersects the circumcircle of the triangle $BCK$ for the second time at point $D$. Prove that $CE$ bisects the angle $BCD$.

2002 Kurschak Competition, 1

Tags: circumcircle , incenter , geometry

We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by $H$, $O$, $I$ respectively. Prove that if a vertex of the triangle lies on the circle $HOI$, then there must be another vertex on this circle as well.

2011 International Zhautykov Olympiad, 3

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

Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)