Olympiad problems

2012 Sharygin Geometry Olympiad, 8

Tags: geometry , incenter

Let $AH$ be an altitude of an acute-angled triangle $ABC$. Points $K$ and $L$ are the projections of $H$ onto sides $AB$ and $AC$. The circumcircle of $ABC$ meets line $KL$ at points $P$ and $Q$, and meets line $AH$ at points $A$ and $T$. Prove that $H$ is the incenter of triangle $PQT$. (M.Plotnikov)

2010 Lithuania National Olympiad, 2

Tags: incenter , circumcircle , inradius , perpendicular bisector , geometry

Let $I$ be the incenter of a triangle $ABC$. $D,E,F$ are the symmetric points of $I$ with respect to $BC,AC,AB$ respectively. Knowing that $D,E,F,B$ are concyclic,find all possible values of $\angle B$.

2014 CentroAmerican, 2

Tags: geometry , incenter , geometric transformation , reflection , trigonometry , circumcircle , angle bisector

Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that \[\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.\]

2011 AMC 12/AHSME, 25

Tags: geometry , incenter , circumcircle , trigonometry , function , inequalities , calculus

Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA \le 90^\circ$, $BC=1$, and $AC \ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of the pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? $\textbf{(A)}\ 60 ^\circ \qquad \textbf{(B)}\ 72 ^\circ\qquad \textbf{(C)}\ 75 ^\circ \qquad \textbf{(D)}\ 80 ^\circ\qquad \textbf{(E)}\ 90 ^\circ$

2011 Turkey Team Selection Test, 1

Tags: incenter , circumcircle , ratio , geometry

Let $D$ be a point different from the vertices on the side $BC$ of a triangle $ABC.$ Let $I, \: I_1$ and $I_2$ be the incenters of the triangles $ABC, \: ABD$ and $ADC,$ respectively. Let $E$ be the second intersection point of the circumcircles of the triangles $AI_1I$ and $ADI_2,$ and $F$ be the second intersection point of the circumcircles of the triangles $AII_2$ and $AI_1D.$ Prove that if $AI_1=AI_2,$ then \[ \frac{EI}{FI} \cdot \frac{ED}{FD}=\frac{{EI_1}^2}{{FI_1}^2}.\]

2009 All-Russian Olympiad, 2

Tags: circumcircle , incenter , ratio , symmetry , geometric transformation , geometry

Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.

2024 Oral Moscow Geometry Olympiad, 2

Tags: geometry , incenter

The bisector $BL$ was drawn in the triangle $ABC$. Let the points $I_1$ and $I_2$ be centers of the circles inscribed in the triangles $ABL$ and $CBL$, and the points $J_1$ and $J_2$ be centers of the excircles of these triangles touching the side $BL$. Prove that the points $I_1$, $I_2$, $J_1$ and $J_2$ lie on the same circle.

2013 Iran Team Selection Test, 1

Tags: incenter , trigonometry , cyclic quadrilateral , geometry

In acute-angled triangle $ABC$, let $H$ be the foot of perpendicular from $A$ to $BC$ and also suppose that $J$ and $I$ are excenters oposite to the side $AH$ in triangles $ABH$ and $ACH$. If $P$ is the point that incircle touches $BC$, prove that $I,J,P,H$ are concyclic.

2013 Princeton University Math Competition, 2

Tags: geometry , incenter , complex number

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) and $I$ be the center of $\gamma$. Let $D$, $E$ and $F$ be the feet of the perpendiculars from $I$ to $BC$, $CA$, and $AB$ respectively. Let $D'$ be the point on $\gamma$ such that $DD'$ is a diameter of $\gamma$. Suppose the tangent to $\gamma$ through $D$ intersects the line $EF$ at $P$. Suppose the tangent to $\gamma$ through $D'$ intersects the line $EF$ at $Q$. Prove that $\angle PIQ + \angle DAD' = 180^{\circ}$.

2005 Taiwan TST Round 3, 2

Tags: incenter , circumcircle , geometry

Given a triangle $ABC$, we construct a circle $\Gamma$ through $B,C$ with center $O$. $\Gamma$ intersects $AC, AB$ at points $D$, $E$, respectively($D$, $E$ are distinct from $B$ and $C$). Let the intersection of $BD$ and $CE$ be $F$. Extend $OF$ so that it intersects the circumcircle of $\triangle ABC$ at $P$. Show that the incenters of triangles $PBD$ and $PCE$ coincide.

2022 Switzerland - Final Round, 8

Tags: geometry , incenter , fixed

Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.

2010 ELMO Shortlist, 3

Tags: incenter , circumcircle , parabola , geometry , conic

A circle $\omega$ not passing through any vertex of $\triangle ABC$ intersects each of the segments $AB$, $BC$, $CA$ in 2 distinct points. Prove that the incenter of $\triangle ABC$ lies inside $\omega$. [i]Evan O' Dorney.[/i]

1994 All-Russian Olympiad, 7

Tags: incenter , geometric transformation , homothety , projective geometry , geometry

Let $ \Gamma_1,\Gamma_2$ and $ \Gamma_3$ be three non-intersecting circles,which are tangent to the circle $ \Gamma$ at points $ A_1,B_1,C_1$,respectively.Suppose that common tangent lines to $ (\Gamma_2,\Gamma_3)$,$ (\Gamma_1,\Gamma_3)$,$ (\Gamma_2,\Gamma_1)$ intersect in points $ A,B,C$. Prove that lines $ AA_1,BB_1,CC_1$ are concurrent.

2020 Iran Team Selection Test, 3

Tags: circumcircle , incenter , geometry

Given a triangle $ABC$ with circumcircle $\Gamma$. Points $E$ and $F$ are the foot of angle bisectors of $B$ and $C$, $I$ is incenter and $K$ is the intersection of $AI$ and $EF$. Suppose that $T$ be the midpoint of arc $BAC$. Circle $\Gamma$ intersects the $A$-median and circumcircle of $AEF$ for the second time at $X$ and $S$. Let $S'$ be the reflection of $S$ across $AI$ and $J$ be the second intersection of circumcircle of $AS'K$ and $AX$. Prove that quadrilateral $TJIX$ is cyclic. [i]Proposed by Alireza Dadgarnia and Amir Parsa Hosseini[/i]

2017 Taiwan TST Round 1, 2

Tags: cyclic quadrilateral , geometry , incenter , reflection , inversion

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2020 Yasinsky Geometry Olympiad, 3

Tags: altitude , geometry , circumcircle , incenter , inradius

The segments $BF$ and $CN$ are the altitudes in the acute-angled triangle $ABC$. The line $OI$, which connects the centers of the circumscribed and inscribed circles of triangle $ABC$, is parallel to the line $FN$. Find the length of the altitude $AK$ in the triangle $ABC$ if the radii of its circumscribed and inscribed circles are $R$ and $r$, respectively. (Grigory Filippovsky)

2012 Centers of Excellency of Suceava, 4

Tags: circumcircle , inradius , incenter , geometry

Let $ O $ be the circumcenter of a triangle $ ABC $ with $ \angle BAC=60^{\circ } $ whose incenter is denoted by $ I. $ Let $ B_1,C_1 $ be the intersection of $ BI,CI $ with the circumcircle of $ ABC, $ respectively. Denote by $ O_1,O_2 $ the circumcenters of $ BIC_1,CIB_1, $ respectively. Show that $ O_1,I,O,O_2 $ are collinear. [i]Cătălin Țigăeru[/i]

2016 India Regional Mathematical Olympiad, 1

Tags: geometry , incenter

Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of triangle $ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$.

2007 AIME Problems, 15

Tags: geometry , inradius , incenter , geometric transformation , homothety , circumcircle , ratio

Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$. If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

1988 IMO Shortlist, 13

Tags: geometry , incenter , ratio , right triangle , area of a triangle

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

2022 Israel TST, 3

Tags: geometry , circumcircle , incenter

Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.

2020 Dutch IMO TST, 1

Tags: geometry , incenter , equal angles

In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.

2025 Benelux, 3

Tags: geometry , incenter , circumcircle

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.

2012 Sharygin Geometry Olympiad, 3

Tags: geometry , centroid , incenter , perpendicular

Let $M$ and $I$ be the centroid and the incenter of a scalene triangle $ABC$, and let $r$ be its inradius. Prove that $MI = r/3$ if and only if $MI$ is perpendicular to one of the sides of the triangle. (A.Karlyuchenko)

2013 Online Math Open Problems, 36

Tags: geometry , trapezoid , incenter , inequalities , pythagorean theorem , triangle inequality

Let $ABCD$ be a nondegenerate isosceles trapezoid with integer side lengths such that $BC \parallel AD$ and $AB=BC=CD$. Given that the distance between the incenters of triangles $ABD$ and $ACD$ is $8!$, determine the number of possible lengths of segment $AD$. [i]Ray Li[/i]