Olympiad problems

2012 Grigore Moisil Intercounty, 3

Let $ \Delta ABC$ be a triangle, with $ m(\angle A)=90^{\circ}$ and $ m(\angle B)=30^{\circ}.$ If $M$ is the middle of $[AB],$ $N$ is the middle of $[BC],$ and $P\in[BC],\ Q\in[MN],$ such that \[\frac{PB}{PC}=4\cdot\frac{QM}{QN}+3,\] prove that $ \Delta APQ$ is an equilateral triangle. [b]Author: MARIN BANCOȘ[/b] [b]Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 24.03.2012, 7th grade[/b]

2001 Canada National Olympiad, 3

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

Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersection point of $XY$ and $BC$. Determine the value of $\frac{BZ}{ZC}$.

2022 Latvia Baltic Way TST, P10

Tags: circumcircle , geometry

Let $\triangle ABC$ be a triangle satisfying $AB<AC$. Let $D$ be a point on the segment $AC$ such that $AB=AD$. Let then $X$ be a point on the segment $BC$ satisfying $BD^2=BX\cdot BC$. Let the circumcircles of the triangles $\triangle XDC$ and $\triangle ABC$ intersect at $M \neq C$. Prove that the line $MD$ goes through the midpoint of the arc $\widehat{BAC}$ of the circumcircle of $\triangle ABC$.

1994 National High School Mathematics League, 3

Tags: geometry , incenter , circumcircle

Circumcircle of $\triangle ABC$ is $\odot O$, incentre of $\triangle ABC$ is $I$. $\angle B=60^{\circ}.\angle A<\angle C$. Bisector of outer angle $\angle A$ intersects $\odot O$ at $E$. Prove: [b](a)[/b] $IO=AE$. [b](b)[/b] The radius of $\odot O$ is $R$, then $2R<IO+IA+IC<(1+\sqrt3)R$.

2010 Paraguay Mathematical Olympiad, 5

Tags: geometry , circumcircle

In a triangle $ABC$, let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively. Let $D'$, $E'$ and $F'$ be the second intersection of lines $AD$, $BE$ and $CF$ with the circumcircle of $ABC$. Show that the triangles $DEF$ and $D'E'F'$ are similar.

1982 IMO Longlists, 13

Tags: trapezoid , geometry , circumcircle , trigonometry , 3d geometry , sphere , pyramid

A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]

2002 Olympic Revenge, 2

Tags: circumcircle , geometry , circles

$ABCD$ is a inscribed quadrilateral. $P$ is the intersection point of its diagonals. $O$ is its circumcenter. $\Gamma$ is the circumcircle of $ABO$. $\Delta$ is the circumcircle of $CDO$. $M$ is the midpoint of arc $AB$ on $\Gamma$ who doesn't contain $O$. $N$ is the midpoint of arc $CD$ on $\Delta$ who doesn't contain $O$. Show that $M,N,P$ are collinear.

2014 Korea - Final Round, 2

Tags: geometry , perpendicular bisector , circumcircle

Let $ABC$ be a isosceles triangle with $ AC = BC > AB$. Let $ E, F $ be the midpoints of segments $ AC, AB$, and let $l$ be the perpendicular bisector of $AC$. Let $ l $ meets $ AB$ at $K$, the line through $B$ parallel to $KC$ meets $AC$ at point $L$, and line $FL$ meets $ l$ at $W$. Let $ P $ be a point on segment $BF$. Let $H$ be the orthocenter of triangle $ACP$ and line $BH$ and $CP$ meet at point $J$. Line $FJ$ meets $l$ at $M$. Prove that $ AW = PW $ if and only if $B$ lies on the circumcircle of $EFM$.

2021 Brazil National Olympiad, 3

Tags: circumcircle , angle bisector , projective geometry , geometry

Let $ABC$ be a scalene triangle and $\omega$ is your incircle. The sides $BC,CA$ and $AB$ are tangents to $\omega$ in $X,Y,Z$ respectively. Let $M$ be the midpoint of $BC$ and $D$ is the intersection point of $BC$ with the angle bisector of $\angle BAC$. Prove that $\angle BAX=\angle MAC$ if and only if $YZ$ passes by the midpoint of $AD$.

2013 NIMO Summer Contest, 12

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

In $\triangle ABC$, $AB = 40$, $BC = 60$, and $CA = 50$. The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$. Find $BP$. [i]Proposed by Eugene Chen[/i]

2015 Benelux, 2

Tags: circumcircle , geometry

Let $ABC$ be an acute triangle with circumcentre $O$. Let $\mathit{\Gamma}_B$ be the circle through $A$ and $B$ that is tangent to $AC$, and let $\mathit{\Gamma}_C$ be the circle through $A$ and $C$ that is tangent to $AB$. An arbitrary line through $A$ intersects $\mathit{\Gamma}_B$ again in $X$ and $\mathit{\Gamma}_C$ again in $Y$. Prove that $|OX|=|OY|$.

2013 ELMO Shortlist, 9

Tags: cyclic quadrilateral , geometry , symmetry , projective geometry , circumcircle , rectangle

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\omega$ whose diagonals meet at $F$. Lines $AB$ and $CD$ meet at $E$. Segment $EF$ intersects $\omega$ at $X$. Lines $BX$ and $CD$ meet at $M$, and lines $CX$ and $AB$ meet at $N$. Prove that $MN$ and $BC$ concur with the tangent to $\omega$ at $X$. [i]Proposed by Allen Liu[/i]

2018 Estonia Team Selection Test, 7

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

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.

2017 Ukrainian Geometry Olympiad, 3

Tags: tangent , geometry , circumcircle , right triangle

On the hypotenuse $AB$ of a right triangle $ABC$, we denote a point $K$ such that $BK = BC$. Let $P$ be a point on the perpendicular from the point $K$ to line $CK$, equidistant from the points $K$ and $B$. Let $L$ be the midpoint of $CK$. Prove that line $AP$ is tangent to the circumcircle of $\Delta BLP$.

2013 Saudi Arabia Pre-TST, 3.4

Tags: geometry , circumcircle , concurrency , incircle

$\vartriangle ABC$ is a triangle with $AB < BC, \Gamma$ its circumcircle, $K$ the midpoint of the minor arc $CA$ of the circle $C$ and $T$ a point on $\Gamma$ such that $KT$ is perpendicular to $BC$. If $A',B'$ are the intouch points of the incircle of $\vartriangle ABC$ with the sides $BC,AC$, prove that the lines $AT,BK,A'B'$ are concurrent.

2012 IMO, 5

Tags: circumcircle , right triangle , geometry

Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$. Show that $MK=ML$. [i]Proposed by Josef Tkadlec, Czech Republic[/i]

2003 Kazakhstan National Olympiad, 6

Tags: circles , circumcircle , geometry

Let the point $ B $ lie on the circle $ S_1 $ and let the point $ A $, other than the point $ B $, lie on the tangent to the circle $ S_1 $ passing through the point $ B $. Let a point $ C $ be chosen outside the circle $ S_1 $, so that the segment $ AC $ intersects $ S_1 $ at two different points. Let the circle $ S_2 $ touch the line $ AC $ at the point $ C $ and the circle $ S_1 $ at the point $ D $, on the opposite side from the point $ B $ with respect to the line $ AC $. Prove that the center of the circumcircle of triangle $ BCD $ lies on the circumcircle of triangle $ ABC $.

2010 APMO, 4

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

Let $ABC$ be an acute angled triangle satisfying the conditions $AB>BC$ and $AC>BC$. Denote by $O$ and $H$ the circumcentre and orthocentre, respectively, of the triangle $ABC.$ Suppose that the circumcircle of the triangle $AHC$ intersects the line $AB$ at $M$ different from $A$, and the circumcircle of the triangle $AHB$ intersects the line $AC$ at $N$ different from $A.$ Prove that the circumcentre of the triangle $MNH$ lies on the line $OH$.

2010 ELMO Shortlist, 6

Tags: circumcircle , geometry

Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic. [i]Carl Lian.[/i]

2013 Sharygin Geometry Olympiad, 14

Tags: trigonometry , analytic geometry , trapezoid , quadratic , asymptote , circumcircle , geometry

Let $M$, $N$ be the midpoints of diagonals $AC$, $BD$ of a right-angled trapezoid $ABCD$ ($\measuredangle A=\measuredangle D = 90^\circ$). The circumcircles of triangles $ABN$, $CDM$ meet the line $BC$ in the points $Q$, $R$. Prove that the distances from $Q$, $R$ to the midpoint of $MN$ are equal.

2019 Estonia Team Selection Test, 2

Tags: geometry , circumcircle

In an acute-angled triangle $ABC$, the altitudes intersect at point $H$, and point $K$ is the foot of the altitude drawn from the vertex $A$. Circle $c$ passing through points $A$ and $K$ intersects sides $AB$ and $AC$ at points $M$ and $N$, respectively. The line passing through point $A$ and parallel to line $BC$ intersects the circumcircles of triangles $AHM$ and $AHN$ for second time, respectively, at points $X$ and $Y$. Prove that $ | X Y | = | BC |$.

2004 South africa National Olympiad, 4

Tags: circumcircle , geometry

Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$) such that $\widehat{A_1CB_1}=\widehat{BAC}$. A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$.

2013 India PRMO, 9

Tags: incenter , geometry , angle , circumcircle

In a triangle $ABC$, let $H, I$ and $O$ be the orthocentre, incentre and circumcentre, respectively. If the points $B, H, I, C$ lie on a circle, what is the magnitude of $\angle BOC$ in degrees?

2012 Iran MO (3rd Round), 4

Tags: circumcircle , angle bisector , college , geometry

The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$. [i]Proposed By Pedram Safaei[/i]

2007 District Olympiad, 1

Tags: equal segments , circumcircle , angle , angle chasing , circumcenter , geometry

Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.