Olympiad problems

2013 Balkan MO, 1

Tags: geometry , incenter , trigonometry , perimeter , circumcircle

In a triangle $ABC$, the excircle $\omega_a$ opposite $A$ touches $AB$ at $P$ and $AC$ at $Q$, while the excircle $\omega_b$ opposite $B$ touches $BA$ at $M$ and $BC$ at $N$. Let $K$ be the projection of $C$ onto $MN$ and let $L$ be the projection of $C$ onto $PQ$. Show that the quadrilateral $MKLP$ is cyclic. ([i]Bulgaria[/i])

2009 Spain Mathematical Olympiad, 2

Tags: incenter , circumcircle , perimeter , geometry

Let $ ABC$ be an acute triangle with the incircle $ C(I,r)$ and the circumcircle $ C(O,R)$ . Denote $ D\in BC$ for which $ AD\perp BC$ and $ AD \equal{} h_a$ . Prove that $ DI^2 \equal{} (2R \minus{} h_a)(h_a \minus{} 2r)$ .

1997 Baltic Way, 15

Tags: incenter , circumcircle , geometry

In the acute triangle $ABC$, the bisectors of $A,B$ and $C$ intersect the circumcircle again at $A_1,B_1$ and $C_1$, respectively. Let $M$ be the point of intersection of $AB$ and $B_1C_1$, and let $N$ be the point of intersection of $BC$ and $A_1B_1$. Prove that $MN$ passes through the incentre of $\triangle ABC$.

V Soros Olympiad 1998 - 99 (Russia), 10.4

Tags: geometry , incenter , angle

Let $M$ be the midpoint of side $BC$ of triangle $ABC$, $Q$ the point of intersection of its angle bisectors. It is known that $MQ=QA$. Find the smallest possible value of angle $\angle MQA$.

2012 Oral Moscow Geometry Olympiad, 1

Tags: geometry , incenter , midpoint

Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?

2019 JBMO Shortlist, G6

Tags: geometry , incenter

Let $ABC$ be a non-isosceles triangle with incenter $I$. Let $D$ be a point on the segment $BC$ such that the circumcircle of $BID$ intersects the segment $AB$ at $E\neq B$, and the circumcircle of $CID$ intersects the segment $AC$ at $F\neq C$. The circumcircle of $DEF$ intersects $AB$ and $AC$ at the second points $M$ and $N$ respectively. Let $P$ be the point of intersection of $IB$ and $DE$, and let $Q$ be the point of intersection of $IC$ and $DF$. Prove that the three lines $EN, FM$ and $PQ$ are parallel. [i]Proposed by Saudi Arabia[/i]

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.

1997 IMO Shortlist, 9

Tags: geometry , incenter , circumcircle , collinearity

Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.

2014 South East Mathematical Olympiad, 3

Tags: incenter , geometry

In an obtuse triangle $ABC$ $(AB>AC)$,$O$ is the circumcentre and $D,E,F$ are the midpoints of $BC,CA,AB$ respectively.Median $AD$ intersects $OF$ and $OE$ at $M$ and $N$ respectively.$BM$ meets $CN$ at point $P$.Prove that $OP\perp AP$

2007 Iran MO (3rd Round), 3

Tags: incenter , circumcircle , trigonometry , geometry

Let $ I$ be incenter of triangle $ ABC$, $ M$ be midpoint of side $ BC$, and $ T$ be the intersection point of $ IM$ with incircle, in such a way that $ I$ is between $ M$ and $ T$. Prove that $ \angle BIM\minus{}\angle CIM\equal{}\frac{3}2(\angle B\minus{}\angle C)$, if and only if $ AT\perp BC$.

2022 Belarusian National Olympiad, 11.6

Tags: geometry , incenter

The incircle of a right-angled triangle $ABC$ touches hypotenus $AB$ at $P$, $BC$ and $AC$ at $R$ and $Q$ respectively. $C_1$ and $C_2$ are reflections of $C$ in $PQ$ and $PR$. Find the angle $C_1IC_2$, where $I$ is the incenter of $ABC$.

2009 India IMO Training Camp, 1

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

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.

2005 Croatia National Olympiad, 2

Tags: geometry , incenter

The lines joining the incenter of a triangle to the vertices divide the triangle into three triangles. If one of these triangles is similar to the initial one,determine the angles of the triangle.

2003 Finnish National High School Mathematics Competition, 1

Tags: incenter , circumcircle , geometry

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

2013 Romania Team Selection Test, 2

Tags: incenter , circumcircle , geometric transformation , homothety , power of a point , geometry

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

2012 Regional Olympiad of Mexico Center Zone, 4

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

On an acute triangle $ABC$ we draw the internal bisector of $<ABC$, $BE$, and the altitude $AD$, ($D$ on $BC$), show that $<CDE$ it's bigger than 45 degrees.

1993 China Team Selection Test, 3

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

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.

2014 Taiwan TST Round 3, 3

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

Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.

2005 ISI B.Stat Entrance Exam, 5

Tags: geometry , incenter , triangle

Consider an acute angled triangle $PQR$ such that $C,I$ and $O$ are the circumcentre, incentre and orthocentre respectively. Suppose $\angle QCR, \angle QIR$ and $\angle QOR$, measured in degrees, are $\alpha, \beta$ and $\gamma$ respectively. Show that \[\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}>\frac{1}{45}\]

2012 India IMO Training Camp, 1

Tags: geometry , incenter , symmetry , reflection

Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$. [i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]

2022 Israel TST, 1

Tags: geometry , incenter

A triangle $ABC$ with orthocenter $H$ is given. $P$ is a variable point on line $BC$. The perpendicular to $BC$ through $P$ meets $BH$, $CH$ at $X$, $Y$ respectively. The line through $H$ parallel to $BC$ meets $AP$ at $Q$. Lines $QX$ and $QY$ meet $BC$ at $U$, $V$ respectively. Find the shape of the locus of the incenters of the triangles $QUV$.

2002 Hungary-Israel Binational, 2

Tags: trigonometry , incenter , inequalities , geometry

Let $A', B' , C'$ be the projections of a point $M$ inside a triangle $ABC$ onto the sides $BC, CA, AB$, respectively. Deﬁne $p(M ) = \frac{MA'\cdot MB'\cdot MC'}{MA \cdot MB \cdot MC}$ . Find the position of point $M$ that maximizes $p(M )$.

2005 District Olympiad, 4

Tags: circumcircle , incenter , ratio , quadratic , angle bisector , perpendicular bisector , geometry

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

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 $\angle BAC=\angle ADC=90^{\circ}$.

2015 Czech-Polish-Slovak Junior Match, 4

Tags: geometry , angle bisector , incenter , right triangle , midpoint

Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .