Olympiad problems

2010 IMO, 4

Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$. [i]Proposed by Marcin E. Kuczma, Poland[/i]

1999 Taiwan National Olympiad, 5

Tags: circumcircle , geometry

Let $AD,BE,CF$ be the altitudes of an acute triangle $ABC$ with $AB>AC$. Line $EF$ meets $BC$ at $P$, and line through $D$ parallel to $EF$ meets $AC$ and $AB$ at $Q$ and $R$, respectively. Let $N$ be any poin on side $BC$ such that $\widehat{NQP}+\widehat{NRP}<180^{0}$. Prove that $BN>CN$.

2002 Italy TST, 1

Tags: circumcircle , geometry

Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.

2017 Thailand TSTST, 3

Tags: circumcircle , equal segments , geometry

In $\vartriangle ABC$ with $AB > AC$, the tangent to the circumcircle at $A$ intersects line $BC$ at $P$. Let $Q$ be the point on $AB$ such that $AQ = AC$, and $A$ lies between $B$ and $Q$. Let $R$ be the point on ray $AP$ such that $AR = CP$. Let $X, Y$ be the midpoints of $AP, CQ$ respectively. Prove that $CR = 2XY$ .

2017 JBMO Shortlist, G3

Tags: geometry , circumcircle , equal angles , angle

Consider triangle $ABC$ such that $AB \le AC$. Point $D$ on the arc $BC$ of thecircumcirle of $ABC$ not containing point $A$ and point $E$ on side $BC$ are such that $\angle BAD = \angle CAE < \frac12 \angle BAC$ . Let $S$ be the midpoint of segment $AD$. If $\angle ADE = \angle ABC - \angle ACB$ prove that $\angle BSC = 2 \angle BAC$ .

2000 239 Open Mathematical Olympiad, 6

Tags: circumcircle , geometry

Let ABCD be a convex quadrilateral, and let M and N be the midpoints of its sides AD and BC, respectively. Assume that the points A, B, M, N are concyclic, and the circumcircle of triangle BMC touches the line AB. Show that the circumcircle of triangle AND touches the line AB, too. Darij

2002 Baltic Way, 13

Tags: circumcircle , parallelogram , geometry

Let $ABC$ be an acute triangle with $\angle BAC>\angle BCA$, and let $D$ be a point on side $AC$ such that $|AB|=|BD|$. Furthermore, let $F$ be a point on the circumcircle of triangle $ABC$ such that line $FD$ is perpendicular to side $BC$ and points $F,B$ lie on different sides of line $AC$. Prove that line $FB$ is perpendicular to side $AC$ .

2010 Peru IMO TST, 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]

2008 Serbia National Math Olympiad, 4

Tags: circumcircle , search , geometry , combinatorics

Each point of a plane is painted in one of three colors. Show that there exists a triangle such that: $ (i)$ all three vertices of the triangle are of the same color; $ (ii)$ the radius of the circumcircle of the triangle is $ 2008$; $ (iii)$ one angle of the triangle is either two or three times greater than one of the other two angles.

2024 Philippine Math Olympiad, P7

Tags: geometry , orthocenter , circumcircle , circumcenter

Let $ABC$ be an acute triangle with orthocenter $H$, circumcenter $O$, and circumcircle $\Omega$. Points $E$ and $F$ are the feet of the altitudes from $B$ to $AC$, and from $C$ to $AB$, respectively. Let line $AH$ intersect $\Omega$ again at $D$. The circumcircle of $DEF$ intersects $\Omega$ again at $X$, and $AX$ intersects $BC$ at $I$. The circumcircle of $IEF$ intersects $BC$ again at $G$. If $M$ is the midpoint of $BC$, prove that lines $MX$ and $OG$ intersect on $\Omega$.

2022 Bulgaria JBMO TST, 4

Tags: geometry , circumcircle

Let $ABC (AC < BC)$ be an acute triangle with circumcircle $k$ and midpoint $P$ of $AB$. The altitudes $AM$ and $BN$ ($M\in BC$, $N\in AC$) intersect at $H$. The point $E$ on $k$ is such that the segments $CE$ and $AB$ are perpendicular. The line $EP$ intersects $k$ again at point $K$ and the point $Q$ on $k$ is such that $KQ$ and $AB$ are parallel. The circumcircle of $AHB$ intersects the segment $CP$ at an interior point $R$. Prove that the points $C$, $M$, $R$, $H$, $N$ and $Q$ are concyclic.

2010 Kyiv Mathematical Festival, 3

Tags: circumcircle , incenter , geometry

Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$.

1980 IMO Shortlist, 4

Tags: geometry , circumcircle , algebra , polynomial , convex polygon

Determine all positive integers $n$ such that the following statement holds: If a convex polygon with with $2n$ sides $A_1 A_2 \ldots A_{2n}$ is inscribed in a circle and $n-1$ of its $n$ pairs of opposite sides are parallel, which means if the pairs of opposite sides \[(A_1 A_2, A_{n+1} A_{n+2}), (A_2 A_3, A_{n+2} A_{n+3}), \ldots , (A_{n-1} A_n, A_{2n-1} A_{2n})\] are parallel, then the sides \[ A_n A_{n+1}, A_{2n} A_1\] are parallel as well.

2017 All-Russian Olympiad, 2

Tags: tangent , geometry , isosceles , circumcircle

Let $ABC$ be an acute angled isosceles triangle with $AB=AC$ and circumcentre $O$. Lines $BO$ and $CO$ intersect $AC, AB$ respectively at $B', C'$. A straight line $l$ is drawn through $C'$ parallel to $AC$. Prove that the line $l$ is tangent to the circumcircle of $\triangle B'OC$.

2008 Regional Competition For Advanced Students, 3

Tags: circumcircle , geometry

Given is an acute angled triangle $ ABC$. Determine all points $ P$ inside the triangle with \[1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2\]

2005 All-Russian Olympiad, 2

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

We have an acute-angled triangle $ABC$, and $AA',BB'$ are its altitudes. A point $D$ is chosen on the arc $ACB$ of the circumcircle of $ABC$. If $P=AA'\cap BD,Q=BB'\cap AD$, show that the midpoint of $PQ$ lies on $A'B'$.

1993 Turkey Team Selection Test, 2

Tags: circumcircle , geometry

Let $M$ be the circumcenter of an acute-angled triangle $ABC$. The circumcircle of triangle $BMA$ intersects $BC$ at $P$ and $AC$ at $Q$. Show that $CM \perp PQ$.

2008 Silk Road, 2

Tags: circumcircle , incenter , angle bisector , geometry

In a triangle $ABC$ $A_0$,$B_0$ and $C_0$ are the midpoints of the sides $BC$,$CA$ and $AB$.$A_1$,$B_1$,$C_1$ are the midpoints of the broken lines $BAC,CAB,ABC$.Show that $A_0A_1,B_0B_1,C_0C_1$ are concurrent.

2009 Brazil Team Selection Test, 4

Tags: geometry , circumcircle , symmetry , homothety , quadrilateral

There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that \[ \angle PAB \plus{} \angle PDC \equal{} \angle PBC \plus{} \angle PAD \equal{} \angle PCD \plus{} \angle PBA \equal{} \angle PDA \plus{} \angle PCB = 90^{\circ} \] if and only if the diagonals $ AC$ and $ BD$ are perpendicular. [i]Proposed by Dusan Djukic, Serbia[/i]

2023 Bulgaria JBMO TST, 3

Tags: geometry , incircle , excircle , similarity , circumcircle , incenter

Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.

2019 Peru IMO TST, 3

Tags: circumcircle , incenter , mixtilinear incircle , geometry

Let $I,\ O$ and $\Gamma$ be the incenter, circumcenter and the circumcircle of triangle $ABC$, respectively. Line $AI$ meets $\Gamma$ at $M$ $(M\neq A)$. The circumference $\omega$ is tangent internally to $\Gamma$ at $T$, and is tangent to the lines $AB$ and $AC$. The tangents through $A$ and $T$ to $\Gamma$ intersect at $P$. Lines $PI$ and $TM$ meet at $Q$. Prove that the lines $QA$ and $MO$ meet at a point on $\Gamma$.

2010 All-Russian Olympiad, 2

Tags: circumcircle , angle bisector , geometric transformation , geometry

Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$. Prove, that $P$, $Q$, $M$ lies at one line.

2001 National Olympiad First Round, 25

Tags: trigonometry , geometry , circumcircle

The circumradius of acute triangle $ABC$ is twice of the distance of its circumcenter to $AB$. If $|AC|=2$ and $|BC|=3$, what is the altitude passing through $C$? $ \textbf{(A)}\ \sqrt {14} \qquad\textbf{(B)}\ \dfrac{3}{7}\sqrt{21} \qquad\textbf{(C)}\ \dfrac{4}{7}\sqrt{21} \qquad\textbf{(D)}\ \dfrac{1}{2}\sqrt{21} \qquad\textbf{(E)}\ \dfrac{2}{3}\sqrt{14} $

1990 IMO Longlists, 90

Tags: incenter , circumcircle , geometry

Let $P$ be a variable point on the circumference of a quarter-circle with radii $OA, OB$ and $\angle AOB = 90^\circ$. H is the projection of $P$ on $OA$. Find the locus of the incenter of the right-angled triangle $HPO.$

2015 Bundeswettbewerb Mathematik Germany, 4

Tags: geometry , incenter , circumcircle , distance , perpendicular

Let $ABC$ be a triangle, such that its incenter $I$ and circumcenter $U$ are distinct. For all points $X$ in the interior of the triangle let $d(X)$ be the sum of distances from $X$ to the three (possibly extended) sides of the triangle. Prove: If two distinct points $P,Q$ in the interior of the triangle $ABC$ satisfy $d(P)=d(Q)$, then $PQ$ is perpendicular to $UI$.