Olympiad problems

2009 Silk Road, 2

Bisectors of triangle ABC of an angles A and C intersect with BC and AB at points A1 and C1 respectively. Lines AA1 and CC1 intersect circumcircle of triangle ABC at points A2 and C2 respectively. K is intersection point of C1A2 and A1C2. I is incenter of ABC. Prove that the line KI divides AC into two equal parts.

1994 APMO, 2

Tags: euler , complex number , analytic geometry , circumcircle , perpendicular bisector , geometry

Given a nondegenerate triangle $ABC$, with circumcentre $O$, orthocentre $H$, and circumradius $R$, prove that $|OH| < 3R$.

2015 Canada National Olympiad, 4

Tags: circumcircle , national olympiads , geometry

Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $I$ be a circle with center on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that \[BP\cdot CQ = AP\cdot AQ.\] Prove that $I$ is tangent to the circumcircle of triangle $BOC$.

2014 Spain Mathematical Olympiad, 3

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

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.

2017 Junior Balkan Team Selection Tests - Romania, 1

Tags: orthocenter , circumcircle , geometry , reflection

Let $P$ be a point in the interior of the acute-angled triangle $ABC$. Prove that if the reflections of $P$ with respect to the sides of the triangle lie on the circumcircle of the triangle, then $P$ is the orthocenter of $ABC$.

2002 Romania National Olympiad, 2

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

Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.

2012 NIMO Problems, 6

Tags: perimeter , rhombus , trigonometry , trig identities , circumcircle , geometry

In rhombus $NIMO$, $MN = 150\sqrt{3}$ and $\measuredangle MON = 60^{\circ}$. Denote by $S$ the locus of points $P$ in the interior of $NIMO$ such that $\angle MPO \cong \angle NPO$. Find the greatest integer not exceeding the perimeter of $S$. [i]Proposed by Evan Chen[/i]

1998 Belarus Team Selection Test, 3

Tags: geometry , circumcircle , collinearity , 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.

2017 China Northern MO, 5

Tags: spiral similarity , circumcircle , geometry

Triangle $ABC$ has $AB > AC$ and $\angle A = 60^\circ $. Let $M$ be the midpoint of $BC$, $N$ be the point on segment $AB$ such that $\angle BNM = 30^\circ$. Let $D,E$ be points on $AB, AC$ respectively. Let $F, G, H$ be the midpoints of $BE, CD, DE$ respectively. Let $O$ be the circumcenter of triangle $FGH$. Prove that $O$ lies on line $MN$.

India EGMO 2021 TST, 3

Tags: circumcircle , geometry , incenter , angle bisector

In acute $\triangle ABC$ with circumcircle $\Gamma$ and incentre $I$, the incircle touches side $AB$ at $F$. The external angle bisector of $\angle ACB$ meets ray $AB$ at $L$. Point $K$ lies on the arc $CB$ of $\Gamma$ not containing $A$, such that $\angle CKI=\angle IKL$. Ray $KI$ meets $\Gamma$ again at $D\ne K$. Prove that $\angle ACF =\angle DCB$.

2021 JHMT HS, 10

Tags: parallelogram , geometry , circumcircle

Parallelogram $JHMT$ satisfies $JH=11$ and $HM=6,$ and point $P$ lies on $\overline{MT}$ such that $JP$ is an altitude of $JHMT.$ The circumcircles of $\triangle{HMP}$ and $\triangle{JMT}$ intersect at the point $Q\neq M.$ Let $A$ be the point lying on $\overline{JH}$ and the circumcircle of $\triangle{JMT}.$ If $MQ=10,$ then the perimeter of $\triangle{JAM}$ can be expressed in the form $\sqrt{a}+\tfrac{b}{c},$ where $a, \ b,$ and $c$ are positive integers, $a$ is not divisible by the square of any prime, and $b$ and $c$ are relatively prime. Find $a+b+c.$

2005 China Second Round Olympiad, 1

Tags: circumcircle , geometry , incenter

In $\triangle ABC$, $AB>AC$, $l$ is a tangent line of the circumscribed circle of $\triangle ABC$, passing through $A$. The circle, centered at $A$ with radius $AC$, intersects $AB$ at $D$, and line $l$ at $E, F$. Prove that lines $DE, DF$ pass through the incenter and an excenter of $\triangle ABC$ respectively.

2002 IMO Shortlist, 1

Tags: incenter , geometry , tangent circle , circumcircle , homothety

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

2020 Korea - Final Round, P5

Tags: geometry , circumcircle

Let $ABC$ be an acute triangle such that $\overline{AB}=\overline{AC}$. Let $M, L, N$ be the midpoints of segment $BC, AM, AC$, respectively. The circumcircle of triangle $AMC$, denoted by $\Omega$, meets segment $AB$ at $P(\neq A)$, and the segment $BL$ at $Q$. Let $O$ be the circumcenter of triangle $BQC$. Suppose that the lines $AC$ and $PQ$ meet at $X$, $OB$ and $LN$ meet at $Y$, and $BQ$ and $CO$ meets at $Z$. Prove that the points $X, Y, Z$ are collinear.

2009 Romania Team Selection Test, 2

Tags: euler , circumcircle , geometry

Prove that the circumcircle of a triangle contains exactly 3 points whose Simson lines are tangent to the triangle's Euler circle and these points are the vertices of an equilateral triangle.

2003 APMO, 2

Tags: rotation , geometry , angle bisector , perimeter , geometric transformation , circumcircle , trigonometry

Suppose $ABCD$ is a square piece of cardboard with side length $a$. On a plane are two parallel lines $\ell_1$ and $\ell_2$, which are also $a$ units apart. The square $ABCD$ is placed on the plane so that sides $AB$ and $AD$ intersect $\ell_1$ at $E$ and $F$ respectively. Also, sides $CB$ and $CD$ intersect $\ell_2$ at $G$ and $H$ respectively. Let the perimeters of $\triangle AEF$ and $\triangle CGH$ be $m_1$ and $m_2$ respectively. Prove that no matter how the square was placed, $m_1+m_2$ remains constant.

2007 Moldova Team Selection Test, 3

Tags: geometry , circumcircle , incenter

Let $ABC$ be a triangle. A circle is tangent to sides $AB, AC$ and to the circumcircle of $ABC$ (internally) at points $P, Q, R$ respectively. Let $S$ be the point where $AR$ meets $PQ$. Show that \[\angle{SBA}\equiv \angle{SCA}\]

2007 Italy TST, 2

Tags: circumcircle , geometry

Let $ABC$ a acute triangle. (a) Find the locus of all the points $P$ such that, calling $O_{a}, O_{b}, O_{c}$ the circumcenters of $PBC$, $PAC$, $PAB$: \[\frac{ O_{a}O_{b}}{AB}= \frac{ O_{b}O_{c}}{BC}=\frac{ O_{c}O_{a}}{CA}\] (b) For all points $P$ of the locus in (a), show that the lines $AO_{a}$, $BO_{b}$ , $CO_{c}$ are cuncurrent (in $X$); (c) Show that the power of $X$ wrt the circumcircle of $ABC$ is: \[-\frac{ a^{2}+b^{2}+c^{2}-5R^{2}}4\] Where $a=BC$ , $b=AC$ and $c=AB$.

2019 CMIMC, 9

Tags: circumcircle , geometry

Let $ABCD$ be a square of side length $1$, and let $P_1, P_2$ and $P_3$ be points on the perimeter such that $\angle P_1P_2P_3 = 90^\circ$ and $P_1, P_2, P_3$ lie on different sides of the square. As these points vary, the locus of the circumcenter of $\triangle P_1P_2P_3$ is a region $\mathcal{R}$; what is the area of $\mathcal{R}$?

2007 Greece Junior Math Olympiad, 1

Tags: angle bisector , circumcircle , incenter , geometry

In a triangle $ABC$ with the incentre $I,$ the angle bisector $AD$ meets the circumcircle of triangle $BIC$ at point $N\neq I$. a) Express the angles of $\triangle BCN$ in terms of the angles of triangle $ABC$. b) Show that the circumcentre of triangle $BIC$ is at the intersection of $AI$ and the circumcentre of $ABC$.

2018 Yasinsky Geometry Olympiad, 3

Tags: construction , geometry , circumcenter , circumcircle , isosceles , angle bisector

Point $O$ is the center of circumcircle $\omega$ of the isosceles triangle $ABC$ ($AB = AC$). Bisector of the angle $\angle C$ intersects $\omega$ at the point $W$. Point $Q$ is the center of the circumcircle of the triangle $OWB$. Construct the triangle $ABC$ given the points $Q,W, B$. (Andrey Mostovy)

2006 Australia National Olympiad, 1

Tags: circumcircle , rhombus , rectangle , analytic geometry , geometry

In a square $ABCD$, $E$ is a point on diagonal $BD$. $P$ and $Q$ are the circumcentres of $\triangle ABE$ and $\triangle ADE$ respectively. Prove that $APEQ$ is a square.

2012 Poland - Second Round, 2

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

Let $ABC$ be a triangle with $\angle A=60^{\circ}$ and $AB\neq AC$, $I$-incenter, $O$-circumcenter. Prove that perpendicular bisector of $AI$, line $OI$ and line $BC$ have a common point.

1994 All-Russian Olympiad, 3

Tags: circumcircle , inequalities , geometry

Let $a,b,c$ be the sides of a triangle, let $m_a,m_b,m_c$ be the corresponding medians, and let $D$ be the diameter of the circumcircle of the triangle. Prove that $\frac{a^2+b^2}{m_c}+\frac{a^2+c^2}{m_b}+\frac{b^2+c^2}{m_a} \leq 6D$.

2009 Hong Kong TST, 4

Tags: ratio , trapezoid , geometric transformation , projective geometry , homothety , circumcircle , geometry

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.