Olympiad problems

2023 Bosnia and Herzegovina Junior BMO TST, 3.

Tags: geometry , incenter

Let ABC be an acute triangle with an incenter $I$.The Incircle touches sides $AC$ and $AB$ in $E$ and $F$ ,respectively. Lines CI and EF intersect at $S$. The point $T$≠$I$ is on the line AI so that $EI$=$ET$.If $K$ is the foot of the altitude from $C$ in triangle $ABC$,prove that points $K$,$S$ and $T$ are colinear.

2008 Finnish National High School Mathematics Competition, 2

Tags: power of a point , radical axis , geometry , incenter , circumcircle

The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.

2017 India IMO Training Camp, 1

Tags: geometry , incenter

Let $ABC$ be an acute angled triangle with incenter $I$. Line perpendicular to $BI$ at $I$ meets $BA$ and $BC$ at points $P$ and $Q$ respectively. Let $D, E$ be the incenters of $\triangle BIA$ and $\triangle BIC$ respectively. Suppose $D,P,Q,E$ lie on a circle. Prove that $AB=BC$.

Kyiv City MO Juniors Round2 2010+ geometry, 2010.89.3

Tags: angle , incenter , geometry

In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles $ABH ,CBH$ respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$.

2006 Iran MO (3rd Round), 3

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

In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that \[LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})\]

2007 All-Russian Olympiad Regional Round, 9.6

Tags: geometry , perpendicular bisector , incenter , circumcircle

Given a triangle. A variable poin $ D$ is chosen on side $ BC$. Points $ K$ and $ L$ are the incenters of triangles $ ABD$ and $ ACD$, respectively. Prove that the second intersection point of the circumcircles of triangles $ BKD$ and $ CLD$ moves along on a fixed circle (while $ D$ moves along segment $ BC$).

2022 China Team Selection Test, 2

Tags: angle bisector , geometry , incenter

Let $ABCD$ be a convex quadrilateral, the incenters of $\triangle ABC$ and $\triangle ADC$ are $I,J$, respectively. It is known that $AC,BD,IJ$ concurrent at a point $P$. The line perpendicular to $BD$ through $P$ intersects with the outer angle bisector of $\angle BAD$ and the outer angle bisector $\angle BCD$ at $E,F$, respectively. Show that $PE=PF$.

2003 German National Olympiad, 2

Tags: midpoint , center , triangle , geometry , circumcircle , incenter

There are four circles $k_1 , k_2 , k_3$ and $k_4$ of equal radius inside the triangle $ABC$. The circle $k_1$ touches the sides $AB, CA$ and the circle $k_4 $, $k_2$ touches the sides $AB,BC$ and $k_4$, and $k_3$ touches the sides $AC, BC$ and $k_4.$ Prove that the center of $k_4$ lies on the line connecting the incenter and circumcenter of $ABC.$

2013 JBMO TST - Turkey, 1

Tags: circumcircle , geometry , incenter

Let $D$ be a point on the side $BC$ of an equilateral triangle $ABC$ where $D$ is different than the vertices. Let $I$ be the excenter of the triangle $ABD$ opposite to the side $AB$ and $J$ be the excenter of the triangle $ACD$ opposite to the side $AC$. Let $E$ be the second intersection point of the circumcircles of triangles $AIB$ and $AJC$. Prove that $A$ is the incenter of the triangle $IEJ$.

2006 IMO, 1

Tags: geometry , incenter , circumcircle

Let $ABC$ be triangle with incenter $I$. A point $P$ in the interior of the triangle satisfies \[\angle PBA+\angle PCA = \angle PBC+\angle PCB.\] Show that $AP \geq AI$, and that equality holds if and only if $P=I$.

2010 Spain Mathematical Olympiad, 2

Tags: incenter , geometry

In a triangle $ABC$, let $P$ be a point on the bisector of $\angle BAC$ and let $A',B'$ and $C'$ be points on lines $BC,CA$ and $AB$ respectively such that $PA'$ is perpendicular to $BC,PB'\perp AC$, and $PC'\perp AB$. Prove that $PA'$ and $B'C'$ intersect on the median $AM$, where $M$ is the midpoint of $BC$.

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 $\Omega$.

2008 India National Olympiad, 1

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

Let $ ABC$ be triangle, $ I$ its in-center; $ A_1,B_1,C_1$ be the reflections of $ I$ in $ BC, CA, AB$ respectively. Suppose the circum-circle of triangle $ A_1B_1C_1$ passes through $ A$. Prove that $ B_1,C_1,I,I_1$ are concylic, where $ I_1$ is the in-center of triangle $ A_1,B_1,C_1$.

2020 Iran Team Selection Test, 3

Tags: circumcircle , geometry , incenter

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]

2016 Stars of Mathematics, 4

Tags: geometry , incenter , circumcircle

Let $ ABC $ be an acute triangle having $ AB<AC, I $ be its incenter, $ D,E,F $ be intersection of the incircle with $ BC, CA, $ respectively, $ AB, X $ be the middle of the arc $ BAC, $ which is an arc of the circumcicle of it, $ P $ be the projection of $ D $ on $ EF $ and $ Q $ be the projection of $ A $ on $ ID. $ [b]a)[/b] Show that $ IX $ and $ PQ $ are parallel. [b]b)[/b] If the circle of diameter $ AI $ intersects the circumcircle of $ ABC $ at $ Y\neq A, $ prove that $ XQ $ intersects $ PI $ at $ Y. $

2014 Sharygin Geometry Olympiad, 4

Tags: incenter , geometry

A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.

2005 ITAMO, 3

Tags: circumcircle , geometry , angle bisector , greatest common divisor , incenter

Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.

2019 Kosovo National Mathematical Olympiad, 3

Tags: incenter , geometry , angle bisector

Let $ABC$ be a triangle with $\angle CAB=60^{\circ}$ and with incenter $I$. Let points $D,E$ be on sides $AC,AB$, respectively, such that $BD$ and $CE$ are angle bisectors of angles $\angle ABC$ and $\angle BCA$, respectively. Show that $ID=IE$.

1993 China Team Selection Test, 3

Tags: reflection , incenter , ring theory , geometry , trigonometry , circumcircle , geometric transformation

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

2021 Sharygin Geometry Olympiad, 7

Tags: geometry , bisects segment , incenter

The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$. Lines $BM$ and $CN$ meet at point $P$. Prove that $PI$ bisects segment $MN$.

1992 IberoAmerican, 3

Tags: geometry , inequalities , circumcircle , trigonometry , inradius , incenter

Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle. a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5. b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.

2010 Chile National Olympiad, 3

Tags: incenter , geometry , midpoint

The sides $BC, CA$, and $AB$ of a triangle $ABC$ are tangent to a circle at points $X, Y, Z$ respectively. Show that the center of such a circle is on the line that passes through the midpoints of $BC$ and $AX$.

2007 All-Russian Olympiad Regional Round, 8.7

Tags: geometry , circumcircle , incenter

Given an isosceles triangle $ ABC$ with $ AB \equal{} BC$. A point $ M$ is chosen inside $ ABC$ such that $ \angle AMC \equal{} 2\angle ABC$ . A point $ K$ lies on segment $ AM$ such that $ \angle BKM \equal{}\angle ABC$. Prove that $ BK \equal{} KM\plus{}MC$.

2010 China Team Selection Test, 1

Tags: trigonometry , circumcircle , power of a point , radical axis , incenter , geometry

Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$. Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.

2017 Philippine MO, 4

Tags: spiral similarity , angle chasing , locus problems , circumcircle , incenter , geometry

Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers at $C_1$ and $C_2$ respectively, intersect at two points $A$ and $B$. Points $P$ and $Q$ are varying points on $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively, such that $P$, $Q$ and $B$ are collinear and $B$ is always between $P$ and $Q$. Let lines $PC_1$ and $QC_2$ intersect at $R$, let $I$ be the incenter of $\Delta PQR$, and let $S$ be the circumcenter of $\Delta PIQ$. Show that as $P$ and $Q$ vary, $S$ traces the arc of a circle whose center is concyclic with $A$, $C_1$ and $C_2$.