Olympiad problems

2010 IMO Shortlist, 6

The vertices $X, Y , Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC, CA, AB$ of an acute-angled triangle $ABC.$ Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ.$ [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

1995 IMO Shortlist, 2

Tags: reflection , circumcircle , complex number , perpendicular bisector , geometry

Let $ A, B$ and $ C$ be non-collinear points. Prove that there is a unique point $ X$ in the plane of $ ABC$ such that \[ XA^2 \plus{} XB^2 \plus{} AB^2 \equal{} XB^2 \plus{} XC^2 \plus{} BC^2 \equal{} XC^2 \plus{} XA^2 \plus{} CA^2.\]

2018 Czech-Polish-Slovak Junior Match, 4

Tags: collinear , circumcenter , circumcircle , geometry

A line passing through the center $M$ of the equilateral triangle $ABC$ intersects sides $BC$ and $CA$, respectively, in points $D$ and $E$. Circumcircles of triangle $AEM$ and $BDM$ intersects, besides point $M$, also at point $P$. Prove that the center of circumcircle of triangle $DEP$ lies on the perpendicular bisector of the segment $AB$.

2011 China Team Selection Test, 1

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

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

2010 Sharygin Geometry Olympiad, 2

Tags: circles , geometry , trigonometry , trigonometric , circumcircle

Each of two equal circles $\omega_1$ and $\omega_2$ passes through the center of the other one. Triangle $ABC$ is inscribed into $\omega_1$, and lines $AC, BC$ touch $\omega_2$ . Prove that $cosA + cosB = 1$.

2006 MOP Homework, 2

Tags: geometry , search , circumcircle

Let $ABC$ be an acute triangle. Determine the locus of points $M$ in the interior of the triangle such that $AB-FG=\frac{MF \cdot AG+MG \cdot BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to lines $BC$ and $AC$, respectively.

2008 IMO Shortlist, 6

Tags: homothety , circumcircle , symmetry , quadrilateral , 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, 9

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

Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$.

2011 Sharygin Geometry Olympiad, 1

Tags: tangent circle , circumcircle , semicircle , altitude , geometry

Altitudes $AA_1$ and $BB_1$ of triangle ABC meet in point $H$. Line $CH$ meets the semicircle with diameter $AB$, passing through $A_1, B_1$, in point $D$. Segments $AD$ and $BB_1$ meet in point $M$, segments $BD$ and $AA_1$ meet in point $N$. Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.

1997 IberoAmerican, 2

Tags: parallelogram , circumcircle , angle bisector , trapezoid , trigonometry , vector , geometry

In an acute triangle $\triangle{ABC}$, let $AE$ and $BF$ be highs of it, and $H$ its orthocenter. The symmetric line of $AE$ with respect to the angle bisector of $\sphericalangle{A}$ and the symmetric line of $BF$ with respect to the angle bisector of $\sphericalangle{B}$ intersect each other on the point $O$. The lines $AE$ and $AO$ intersect again the circuncircle to $\triangle{ABC}$ on the points $M$ and $N$ respectively. Let $P$ be the intersection of $BC$ with $HN$; $R$ the intersection of $BC$ with $OM$; and $S$ the intersection of $HR$ with $OP$. Show that $AHSO$ is a paralelogram.

2009 Croatia Team Selection Test, 3

Tags: circumcircle , geometry , incenter

A triangle $ ABC$ is given with $ \left|AB\right| > \left|AC\right|$. Line $ l$ tangents in a point $ A$ the circumcirle of $ ABC$. A circle centered in $ A$ with radius $ \left|AC\right|$ cuts $ AB$ in the point $ D$ and the line $ l$ in points $ E, F$ (such that $ C$ and $ E$ are in the same halfplane with respect to $ AB$). Prove that the line $ DE$ passes through the incenter of $ ABC$.

2007 Turkey MO (2nd round), 1

Tags: geometry , circumcircle

In an acute triangle $ABC$, the circle with diameter $AC$ intersects $AB$ and $AC$ at $K$ and $L$ different from $A$ and $C$ respectively. The circumcircle of $ABC$ intersects the line $CK$ at the point $F$ different from $C$ and the line $AL$ at the point $D$ different from $A$. A point $E$ is choosen on the smaller arc of $AC$ of the circumcircle of $ABC$ . Let $N$ be the intersection of the lines $BE$ and $AC$ . If $AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}$ prove that $\angle KNB= \angle BNL$ .

2014 Oral Moscow Geometry Olympiad, 3

Tags: concurrency , circumcircle , angle bisector , geometry

The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.

2021 Iran Team Selection Test, 5

Tags: trapezoid , angle bisector , circumcircle , geometry

Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$. Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$. Proposed by [i]Alireza Dadgarnia[/i]

2006 Polish MO Finals, 3

Tags: geometry , circumcircle , reflection , rhombus , congruent triangles , geometric transformation , perpendicular bisector

Let $ABCDEF$ be a convex hexagon satisfying $AC=DF$, $CE=FB$ and $EA=BD$. Prove that the lines connecting the midpoints of opposite sides of the hexagon $ABCDEF$ intersect in one point.

2022 Dutch IMO TST, 1

Tags: circumcircle , geometry

Consider an acute triangle $ABC$ with $|AB| > |CA| > |BC|$. The vertices $D, E$, and $F$ are the base points of the altitudes from $A, B$, and $C$, respectively. The line through F parallel to $DE$ intersects $BC$ in $M$. The angular bisector of $\angle MF E$ intersects $DE$ in $N$. Prove that $F$ is the circumcentre of $\vartriangle DMN$ if and only if $B$ is the circumcentre of $\vartriangle FMN$.

2020 Tuymaada Olympiad, 6

Tags: geometry , triangle , circumcircle

An isosceles triangle $ABC$ ($AB = BC$) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$. The side $AC$ meets the circles again at points $Y$ and $Z$. $O$ is the circumcenter of the triangle $XYZ$. Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$.

2001 China Team Selection Test, 2

Tags: circumcircle , geometry , incenter

In the equilateral $\bigtriangleup ABC$, $D$ is a point on side $BC$. $O_1$ and $I_1$ are the circumcenter and incenter of $\bigtriangleup ABD$ respectively, and $O_2$ and $I_2$ are the circumcenter and incenter of $\bigtriangleup ADC$ respectively. $O_1I_1$ intersects $O_2I_2$ at $P$. Find the locus of point $P$ as $D$ moves along $BC$.

2006 Spain Mathematical Olympiad, 3

Tags: circumcircle , isosceles , distance , geometry

$ABC$ is an isosceles triangle with $AB = AC$. Let $P$ be any point of a circle tangent to the sides $AB$ in $B$ and to AC in C. Denote $a$, $b$ and $c$ to the distances from $P$ to the sides $BC, AC$ and $AB$ respectively. Prove that: $a^2=bc$

2013 Romania National Olympiad, 3

Tags: geometry , perimeter , analytic geometry , inradius , circumcircle

Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$ a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$ b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$

1935 Moscow Mathematical Olympiad, 011

Tags: geometry , circumradius , radius , circumcircle

In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ , $BC$ and intersect $BC$ , $AC$ at $F$ , $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.

2016 ELMO Problems, 6

Tags: circumcircle , incircle , inversion , tangent circle , angle bisector , geometry

Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$, let its incircle be tangent to sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$, respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$. Finally, let $\gamma$ be the circumcircle of $\triangle AST$. (a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$. (b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$. [i]James Lin[/i]

2017 Azerbaijan Team Selection Test, 1

Tags: circumcircle , geometry

Let $ABC$ be an acute angled triangle. Points $E$ and $F$ are chosen on the sides $AC$ and $AB$, respectively, such that \[BC^2=BA\times BF+CE\times CA.\] Prove that for all such $E$ and $F$, circumcircle of the triangle $AEF$ passes through a fixed point different from $A$.

2005 Alexandru Myller, 2

Tags: geometry , circumcircle

Let $ ABC $ be a triangle with $ \angle BAC <90^{\circ } . $ In the exterior of $ ABC, $ choose the points $ D,E $ such that $ DA=DB,EA=EC $ and $ \angle ADB =\angle AEC =2\angle BAC . $ Show that the symmetric of $ A $ with respect to the midpoint of the segment $ DE $ is the circumcircle of $ ABC. $

2011 NIMO Problems, 5

Tags: circumcircle , perimeter , perpendicular bisector , geometry

In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle. [i]Proposed by Isabella Grabski [/i]