Olympiad problems

1986 National High School Mathematics League, 6

Tags: circumcircle , geometry

Area of $\triangle ABC$ is $\frac{1}{4}$, circumradius of $\triangle ABC$ is $1$. Let $s=\sqrt{a}+\sqrt{b}+\sqrt{c},t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, then $\text{(A)}s>t\qquad\text{(B)}s=t\qquad\text{(C)}s<t\qquad\text{(D)}s>t$

1997 Taiwan National Olympiad, 8

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

Let $O$ be the circumcenter and $R$ be the circumradius of an acute triangle $ABC$. Let $AO$ meet the circumcircle of $OBC$ again at $D$, $BO$ meet the circumcircle of $OCA$ again at $E$, and $CO$ meet the circumcircle of $OAB$ again at $F$. Show that $OD.OE.OF\geq 8R^{3}$.

2016 Regional Olympiad of Mexico Southeast, 6

Tags: geometry , circumcircle

Let $M$ the midpoint of $AC$ of an acutangle triangle $ABC$ with $AB>BC$. Let $\Omega$ the circumcircle of $ABC$. Let $P$ the intersection of the tangents to $\Omega$ in point $A$ and $C$ and $S$ the intersection of $BP$ and $AC$. Let $AD$ the altitude of triangle $ABP$ with $D$ in $BP$ and $\omega$ the circumcircle of triangle $CSD$. Let $K$ and $C$ the intersections of $\omega$ and $\Omega (K\neq C)$. Prove that $\angle CKM=90^\circ$.

2015 Cono Sur Olympiad, 3

Tags: geometry , circumcircle

Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined: $C_n$ the foot of the perpendicular from $P$ to $A_nB_n$ $O_n$ is the center of $\odot (PA_nB_n)$ $A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$ $B_{n+1} \equiv PB_n \cap O_nA_{n+1}$ If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$ [hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]

2002 Finnish National High School Mathematics Competition, 5

Tags: geometry , circumcircle , modular arithmetic , combinatorics

There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$ What is the least number of colours which suffices?

2007 Peru IMO TST, 2

Tags: analytic geometry , geometry , trigonometry , circumcircle , ratio , rectangle , geometric transformation

Let $ABC$ be a triangle such that $CA \neq CB$, the points $A_{1}$ and $B_{1}$ are tangency points for the ex-circles relative to sides $CB$ and $CA$, respectively, and $I$ the incircle. The line $CI$ intersects the cincumcircle of the triangle $ABC$ in the point $P$. The line that trough $P$ that is perpendicular to $CP$, intersects the line $AB$ in $Q$. Prove that the lines $QI$ and $A_{1}B_{1}$ are parallels.

2023 Dutch BxMO TST, 4

Tags: circumcircle , geometry

In a triangle $\triangle ABC$ with $\angle ABC < \angle BCA$, we define $K$ as the excenter with respect to $A$. The lines $AK$ and $BC$ intersect in a point $D$. Let $E$ be the circumcenter of $\triangle BKC$. Prove that \[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]

2009 Brazil Team Selection Test, 4

Tags: symmetry , circumcircle , quadrilateral , homothety , geometry

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]

2013 Sharygin Geometry Olympiad, 4

Tags: circumcircle , geometry , trapezoid , perpendicular bisector

Let $ABC$ be a nonisosceles triangle. Point $O$ is its circumcenter, and point $K$ is the center of the circumcircle $w$ of triangle $BCO$. The altitude of $ABC$ from $A$ meets $w$ at a point $P$. The line $PK$ intersects the circumcircle of $ABC$ at points $E$ and $F$. Prove that one of the segments $EP$ and $FP$ is equal to the segment $PA$.

2019 Taiwan TST Round 3, 6

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

Given a triangle $ \triangle{ABC} $ with circumcircle $ \Omega $. Denote its incenter and $ A $-excenter by $ I, J $, respectively. Let $ T $ be the reflection of $ J $ w.r.t $ BC $ and $ P $ is the intersection of $ BC $ and $ AT $. If the circumcircle of $ \triangle{AIP} $ intersects $ BC $ at $ X \neq P $ and there is a point $ Y \neq A $ on $ \Omega $ such that $ IA = IY $. Show that $ \odot\left(IXY\right) $ tangents to the line $ AI $.

2019 Iran MO (3rd Round), 2

Tags: geometry , circumcircle

In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.

2010 APMO, 1

Tags: circumcircle , perpendicular bisector , geometry

Let $ABC$ be a triangle with $\angle BAC \neq 90^{\circ}.$ Let $O$ be the circumcenter of the triangle $ABC$ and $\Gamma$ be the circumcircle of the triangle $BOC.$ Suppose that $\Gamma$ intersects the line segment $AB$ at $P$ different from $B$, and the line segment $AC$ at $Q$ different from $C.$ Let $ON$ be the diameter of the circle $\Gamma.$ Prove that the quadrilateral $APNQ$ is a parallelogram.

2011 Akdeniz University MO, 4

Tags: geometry , circumcircle

Let an acute-angled triangle $ABC$'s circumcircle is $S$. $S$'s tangent from $B$ and $C$ intersects at point $M$. A line, lies $M$ and parallel to $[AB]$ intersects with $S$ at points $D$ and $E$, intersect with $[AC]$ at point $F$. Prove that $$[DF]=[FE]$$

2018 Estonia Team Selection Test, 7

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

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.

2004 National Olympiad First Round, 29

Tags: trigonometry , circumcircle , law of cosines , cyclic quadrilateral , trapezoid , geometry

Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral? $ \textbf{(A)}\ 5\sqrt 3 \qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{34} $

2005 IMO Shortlist, 1

Tags: circumcircle , triangle -incenter , geometry , homothety

Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

2008 IMO Shortlist, 2

Tags: circumcircle , geometry , trapezoid

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

1983 IMO Longlists, 74

Tags: conic , geometry , circumcircle , hyperbola , asymptote , perpendicular bisector , analytic geometry

In a plane we are given two distinct points $A,B$ and two lines $a, b$ passing through $B$ and $A$ respectively $(a \ni B, b \ni A)$ such that the line $AB$ is equally inclined to a and b. Find the locus of points $M$ in the plane such that the product of distances from $M$ to $A$ and a equals the product of distances from $M$ to $B$ and $b$ (i.e., $MA \cdot MA' = MB \cdot MB'$, where $A'$ and $B'$ are the feet of the perpendiculars from $M$ to $a$ and $b$ respectively).

2009 Junior Balkan MO, 4

Tags: analytic geometry , ceiling function , geometry , circumcircle , combinatorics

Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.

2014 Iran Team Selection Test, 1

Tags: geometry , circumcircle

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

2013 Baltic Way, 12

Tags: trapezoid , circumcircle , geometry

A trapezoid $ABCD$ with bases $AB$ and $CD$ is such that the circumcircle of the triangle $BCD$ intersects the line $AD$ in a point $E$, distinct from $A$ and $D$. Prove that the circumcircle oF the triangle $ABE$ is tangent to the line $BC$.

2020 Kosovo National Mathematical Olympiad, 3

Tags: concyclic , circumcenter , circumcircle , geometry

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$. The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$. Show that points $A$, $O$, $P$ and $Q$ are concyclic.

2017 Iberoamerican, 4

Tags: circumcircle , concurrency , geometry

Let $ABC$ be an acute triangle with $AC > AB$ and $O$ its circumcenter. Let $D$ be a point on segment $BC$ such that $O$ lies inside triangle $ADC$ and $\angle DAO + \angle ADB = \angle ADC$. Let $P$ and $Q$ be the circumcenters of triangles $ABD$ and $ACD$ respectively, and let $M$ be the intersection of lines $BP$ and $CQ$. Show that lines $AM, PQ$ and $BC$ are concurrent. [i]Pablo Jaén, Panama[/i]

1980 IMO Shortlist, 4

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

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.

2011 China Second Round Olympiad, 1

Tags: geometry , symmetry , circumcircle , cyclic quadrilateral

Let $P,Q$ be the midpoints of diagonals $AC,BD$ in cyclic quadrilateral $ABCD$. If $\angle BPA=\angle DPA$, prove that $\angle AQB=\angle CQB$.