Olympiad problems

2010 Iran MO (3rd Round), 4

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

in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)

2014 Online Math Open Problems, 19

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

In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, determine $m+n$. [i]Proposed by Ray Li[/i]

2017 Sharygin Geometry Olympiad, P5

Tags: locus , acute triangle , incenter , geometry

A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of а) the vertices of their greatest angles, b) their incenters.

2013 IMAC Arhimede, 3

Tags: geometry , angle bisector , concyclic , incenter

Let $ABC$ be a triangle with $\angle ABC=120^o$ and triangle bisectors $(AA_1),(BB_1),(CC_1)$, respectively. $B_1F \perp A_1C_1$, where $F\in (A_1C_1)$. Let $R,I$ and $S$ be the centers of the circles which are inscribed in triangles $C_1B_1F,C_1B_1A_1, A_1B_1F$, and $B_1S\cap A_1C_1=\{Q\}$. Show that $R,I,S,Q$ are on the same circle.

1986 IMO Shortlist, 14

Tags: collinearity , circumcircle , incenter , geometry

The circle inscribed in a triangle $ABC$ touches the sides $BC,CA,AB$ in $D,E, F$, respectively, and $X, Y,Z$ are the midpoints of $EF, FD,DE$, respectively. Prove that the centers of the inscribed circle and of the circles around $XYZ$ and $ABC$ are collinear.

1999 All-Russian Olympiad, 3

Tags: geometry , incenter

A triangle $ABC$ is inscribed in a circle $S$. Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively. The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$. Prove that the incenter $I$ of $\triangle ABC$ lies on a common tangent to $S_1$ and $S_2$.

2013 China Team Selection Test, 2

Tags: geometry , circumcircle , incenter , trigonometry

The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively. Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$

2014 Tuymaada Olympiad, 7

Tags: incenter , geometry , analytic geometry , parallelogram

A parallelogram $ABCD$ is given. The excircle of triangle $\triangle{ABC}$ touches the sides $AB$ at $L$ and the extension of $BC$ at $K$. The line $DK$ meets the diagonal $AC$ at point $X$; the line $BX$ meets the median $CC_1$ of trianlge $\triangle{ABC}$ at ${Y}$. Prove that the line $YL$, median $BB_1$ of triangle $\triangle{ABC}$ and its bisector $CC^\prime$ have a common point. [i](A. Golovanov)[/i]

2009 China Team Selection Test, 1

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

Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$

2016 Ukraine Team Selection Test, 8

Tags: incenter , parallel , circumcircle , geometry , incircle

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

JBMO Geometry Collection, 2007

Tags: function , angle bisector , incenter , circumcircle , cyclic quadrilateral , geometry , trigonometry

Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$ , $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$ . Determine the measure of $\angle{APD}$.

2010 Poland - Second Round, 2

Tags: incenter , 3d geometry , geometry , tetrahedron , circumcircle

The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

1997 IMO Shortlist, 25

Tags: incenter , parallelogram , geometry , concurrency , circumcircle

Let $ X,Y,Z$ be the midpoints of the small arcs $ BC,CA,AB$ respectively (arcs of the circumcircle of $ ABC$). $ M$ is an arbitrary point on $ BC$, and the parallels through $ M$ to the internal bisectors of $ \angle B,\angle C$ cut the external bisectors of $ \angle C,\angle B$ in $ N,P$ respectively. Show that $ XM,YN,ZP$ concur.

1997 Romania Team Selection Test, 4

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

Let $ABC$ be a triangle, $D$ be a point on side $BC$, and let $\mathcal{O}$ be the circumcircle of triangle $ABC$. Show that the circles tangent to $\mathcal{O},AD,BD$ and to $\mathcal{O},AD,DC$ are tangent to each other if and only if $\angle BAD=\angle CAD$. [i]Dan Branzei[/i]

2001 India National Olympiad, 1

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

Let $ABC$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_1, B_1, C_1$ denote the reflections of $P$ in the sides $BC,CA,AB$ respectively. Prove that (i) If $P$ is the incenter or an excentre of $ABC$, then $P$ is the circumenter of $A_1B_1C_1$; (ii) If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_1B_1C_1$; (iii) If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_1B_1C_1$.

2013 Taiwan TST Round 1, 6

Tags: incenter , geometry , homothety

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

2009 Germany Team Selection Test, 2

Tags: circumcircle , geometry , incenter

Let triangle $ABC$ be perpendicular at $A.$ Let $M$ be the midpoint of segment $\overline{BC}.$ Point $D$ lies on side $\overline{AC}$ and satisfies $|AD|=|AM|.$ Let $P \neq C$ be the intersection of the circumcircle of triangles $AMC$ and $BDC.$ Prove that $CP$ bisects the angle at $C$ of triangle $ABC.$

1997 Akdeniz University MO, 5

Tags: perimeter , geometry , incenter

An $ABC$ triangle divide by a $d$ line such that, new two pieces' areas and perimeters are equal. Prove that $ABC$'s incenter lies $d$

Cono Sur Shortlist - geometry, 1993.9

Tags: bisector , perimeter , incenter , area , geometry

Prove that a line that divides a triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. Prove an analogous property for a polygon that has an inscribed circle.

2014 Saint Petersburg Mathematical Olympiad, 6

Tags: geometry , incenter

Points $A,B$ are on circle $\omega$. Points $C$ and $D$ are moved on the arc $AB$, such that $CD$ has constant length. $I_1,I_2$ - incenters of $ABC$ and $ABD$. Prove that line $I_1I_2$ is tangent to some fixed circle.

2018 Saudi Arabia BMO TST, 4

Tags: orthocenter , perpendicular , circles , incenter , geometry

Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively. a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$. b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.

1988 Federal Competition For Advanced Students, P2, 5

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

The bisectors of angles $ B$ and $ C$ of triangle $ ABC$ intersect the opposite sides in points $ B'$ and $ C'$ respectively. Show that the line $ B'C'$ intersects the incircle of the triangle.

2008 Bulgarian Autumn Math Competition, Problem 12.2

Tags: excircle , collinear points , incenter , geometry

Let $ABC$ be a triangle, such that the midpoint of $AB$, the incenter and the touchpoint of the excircle opposite $A$ with $\overline{AC}$ are collinear. Find $AB$ and $BC$ if $AC=3$ and $\angle ABC=60^{\circ}$.

2019 Olympic Revenge, 3

Tags: construction , geometry , incenter , euler line

Let $\Gamma$ be a circle centered at $O$ with radius $R$. Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$. Let $I$ be a point such that $IX = IY$ and $XY = OI$. Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$, incenter $I$ and Euler line $OX$. Prove that this triangle is unique.

2022 JHMT HS, 10

Tags: circumcircle , geometry , incenter

In $\triangle JMT$, $JM=410$, $JT=49$, and $\angle{MJT}>90^\circ$. Let $I$ and $H$ be the incenter and orthocenter of $\triangle JMT$, respectively. The circumcircle of $\triangle JIH$ intersects $\overleftrightarrow{JT}$ at a point $P\neq J$, and $IH=HP$. Find $MT$.