Olympiad problems

2023-24 IOQM India, 23

In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$. Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin $(0,0)$.

2010 ELMO Shortlist, 4

Tags: circumcircle , incenter , homothety , inversion , spiral similarity , geometry

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]

2013 Kazakhstan National Olympiad, 2

Tags: incenter , circumcircle , geometric transformation , geometry

Given triangle ABC with incenter I. Let P,Q be point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.

1978 Czech and Slovak Olympiad III A, 5

Tags: trapezoid , incenter , rectangle , isosceles trapezoid , geometry

Let $ABCS$ be an isosceles trapezoid. Denote $A',B',C',D'$ the incenters of triangles $BCD,CDA,$ $DAB,ABC,$ respectively. Show that $A',B',C',D'$ are vertices of a rectangle.

2015 India IMO Training Camp, 1

Tags: geometry , reflection , incenter

In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.

2010 Serbia National Math Olympiad, 1

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

Let $O$ be the circumcenter of triangle $ABC$. A line through $O$ intersects the sides $CA$ and $CB$ at points $D$ and $E$ respectively, and meets the circumcircle of $ABO$ again at point $P \neq O$ inside the triangle. A point $Q$ on side $AB$ is such that $\frac{AQ}{QB}=\frac{DP}{PE}$. Prove that $\angle APQ = 2\angle CAP$. [i]Proposed by Dusan Djukic[/i]

2004 Silk Road, 3

Tags: circumcircle , incenter , trigonometry , geometry

In-circle of $ABC$ with center $I$ touch $AB$ and $AC$ at $P$ and $Q$ respectively. $BI$ and $CI$ intersect $PQ$ at $K$ and $L$ respectively. Prove, that circumcircle of $ILK$ touch incircle of $ABC$ iff $|AB|+|AC|=3|BC|$.

Kyiv City MO Juniors Round2 2010+ geometry, 2012.8.5

Tags: incenter , equilateral , angle , geometry

In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$. (Rozhkova Maria)

2006 IMO Shortlist, 4

Tags: incenter , triangle , geometry , midpoint , congruent triangles

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.

2022 Junior Balkan Team Selection Tests - Moldova, 8

Tags: incenter , angle , geometry

Let $ABC$ be the triangle and $I$ the center of the circle inscribed in this triangle. The point $M$, located on the tangent taken to the point $B$ to the circumscribed circle of the triangle $ABC$, satisfies the relation $AB = MB$. Point $N$, located on the tangent taken to point $C$ to the same circle, satisfies the relation $AC = NC$. Points $M, A$ and $N$ lie on the same side of the line $BC$. Prove that $$\angle BAC + \angle MIN = 180^o.$$

2014 EGMO, 2

Tags: incenter , circumcircle , reflection , homothety , inversion

Let $D$ and $E$ be points in the interiors of sides $AB$ and $AC$, respectively, of a triangle $ABC$, such that $DB = BC = CE$. Let the lines $CD$ and $BE$ meet at $F$. Prove that the incentre $I$ of triangle $ABC$, the orthocentre $H$ of triangle $DEF$ and the midpoint $M$ of the arc $BAC$ of the circumcircle of triangle $ABC$ are collinear.

2001 AIME Problems, 7

Tags: inradius , incenter , analytic geometry , similar triangles , geometry

Let $\triangle{PQR}$ be a right triangle with $PQ=90$, $PR=120$, and $QR=150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{10n}$. What is $n$?

Estonia Open Senior - geometry, 2014.1.4

Tags: geometry , incenter , equilateral

In a plane there is a triangle $ABC$. Line $AC$ is tangent to circle $c_A$ at point $C$ and circle $c_A$ passes through point $B$. Line $BC$ is tangent to circle $c_B$ at point $C$ and circle $c_B$ passes through point $A$. The second intersection point $S$ of circles $c_A$ and $c_B$ coincides with the incenter of triangle $ABC$. Prove that the triangle $ABC$ is equilateral.

Kyiv City MO Seniors 2003+ geometry, 2020.10.5

Tags: geometry , euler line , incenter , collinear , perpendicular , isosceles

Given an acute isosceles triangle $ABC, AK$ and $CN$ are its angle bisectors, $I$ is their intersection point . Let point $X$ be the other intersection point of the circles circumscribed around $\vartriangle ABC$ and $\vartriangle KBN$. Let $M$ be the midpoint of $AC$. Prove that the Euler line of $\vartriangle ABC$ is perpendicular to the line $BI$ if and only if the points $X, I$ and $M$ lie on the same line. (Kivva Bogdan)

2011 Belarus Team Selection Test, 1

Tags: geometry , incenter , orthocenter , perpendicular

In an acute-angled triangle $ABC$, the orthocenter is $H$. $I_H$ is the incenter of $\vartriangle BHC$. The bisector of $\angle BAC$ intersects the perpendicular from $I_H$ to the side $BC$ at point $K$. Let $F$ be the foot of the perpendicular from $K$ to $AB$. Prove that $2KF+BC=BH +HC$ A. Voidelevich

2014 NIMO Problems, 14

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

Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$. Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

2022 Novosibirsk Oral Olympiad in Geometry, 4

Tags: incenter , circumcircle , geometry

A point $D$ is marked on the side $AC$ of triangle $ABC$. The circumscribed circle of triangle $ABD$ passes through the center of the inscribed circle of triangle $BCD$. Find $\angle ACB$ if $\angle ABC = 40^o$.

2016 Saudi Arabia BMO TST, 2

Tags: incenter , collinear , circumcircle , tangent circle , geometry

Let $ABC$ be a triangle and $I$ its incenter. The point $D$ is on segment $BC$ and the circle $\omega$ is tangent to the circumcirle of triangle $ABC$ but is also tangent to $DC, DA$ at $E, F$, respectively. Prove that $E, F$ and $I$ are collinear.

2010 Poland - Second Round, 1

Tags: incenter , perpendicular bisector , angle bisector , geometry

In the convex pentagon $ABCDE$ all interior angles have the same measure. Prove that the perpendicular bisector of segment $EA$, the perpendicular bisector of segment $BC$ and the angle bisector of $\angle CDE$ intersect in one point.

2014 Sharygin Geometry Olympiad, 18

Tags: geometry , circumcircle , incenter

Let $I$ be the incenter of a circumscribed quadrilateral $ABCD$. The tangents to circle $AIC$ at points $A, C$ meet at point $X$. The tangents to circle $BID$ at points $B, D$ meet at point $Y$ . Prove that $X, I, Y$ are collinear.

Indonesia MO Shortlist - geometry, g10

Tags: excenter , geometry , incenter , excircle

Given a triangle $ABC$ with incenter $I$ . It is known that $E_A$ is center of the ex-circle tangent to $BC$. Likewise, $E_B$ and $E_C$ are the centers of the ex-circles tangent to $AC$ and $AB$, respectively. Prove that $I$ is the orthocenter of the triangle $E_AE_BE_C$.

2022 Austrian MO Regional Competition, 3

Tags: geometry , incenter , concurrency

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$. [i](Karl Czakler)[/i]

1996 Balkan MO, 1

Tags: geometry , circumcircle , incenter , euler , inequalities

Let $O$ be the circumcenter and $G$ be the centroid of a triangle $ABC$. If $R$ and $r$ are the circumcenter and incenter of the triangle, respectively, prove that \[ OG \leq \sqrt{ R ( R - 2r ) } . \] [i]Greece[/i]

1997 IMO Shortlist, 16

Tags: geometry , incenter , circumcircle , trigonometry , angle bisector , collinearity

In an acute-angled triangle $ ABC,$ let $ AD,BE$ be altitudes and $ AP,BQ$ internal bisectors. Denote by $ I$ and $ O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $ D, E,$ and $ I$ are collinear if and only if the points $ P, Q,$ and $ O$ are collinear.

2007 France Team Selection Test, 3

Tags: incenter , triangle , midpoint , congruent triangles , geometry

A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.