Olympiad problems

2007 Iran MO (3rd Round), 3

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$.

Russian TST 2017, P1

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 $BDE$.

2018 Saudi Arabia BMO TST, 4

Tags: geometry , circles , orthocenter , perpendicular , incenter

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 $H$.

2013 ELMO Shortlist, 8

Tags: geometry , circumcircle , reflection , incenter

Let $ABC$ be a triangle, and let $D$, $A$, $B$, $E$ be points on line $AB$, in that order, such that $AC=AD$ and $BE=BC$. Let $\omega_1, \omega_2$ be the circumcircles of $\triangle ABC$ and $\triangle CDE$, respectively, which meet at a point $F \neq C$. If the tangent to $\omega_2$ at $F$ cuts $\omega_1$ again at $G$, and the foot of the altitude from $G$ to $FC$ is $H$, prove that $\angle AGH=\angle BGH$. [i]Proposed by David Stoner[/i]

2002 Croatia Team Selection Test, 2

Tags: geometry , incenter , equal segments

A quadrilateral $ABCD$ is circumscribed about a circle. Lines $AC$ and $DC$ meet at point $E$ and lines $DA$ and $BC$ meet at $F$, where $B$ is between $A$ and $E$ and between $C$ and $F$. Let $I_1, I_2$ and $I_3$ be the incenters of triangles $AFB, BEC$ and $ABC$, respectively. The line $I_1I_3$ intersects $EA$ at $K$ and $ED$ at $L$, whereas the line $I_2I_3$ intersects $FC$ at $M$ and $FD$ at $N$. Prove that $EK = EL$ if and only if $FM = FN$

2004 Turkey Team Selection Test, 2

Tags: geometry , circumcircle , incenter , geometric transformation , reflection , projective geometry , perpendicular bisector

Let $\triangle ABC$ be an acute triangle, $O$ be its circumcenter, and $D$ be a point different that $A$ and $C$ on the smaller $AC$ arc of its circumcircle. Let $P$ be a point on $[AB]$ satisfying $\widehat{ADP} = \widehat {OBC}$ and $Q$ be a point on $[BC]$ satisfying $\widehat{CDQ}=\widehat {OBA}$. Show that $\widehat {DPQ} = \widehat {DOC}$.

2010 Federal Competition For Advanced Students, Part 1, 4

Tags: parallelogram , incenter , perimeter , geometry

The the parallel lines through an inner point $P$ of triangle $\triangle ABC$ split the triangle into three parallelograms and three triangles adjacent to the sides of $\triangle ABC$. (a) Show that if $P$ is the incenter, the perimeter of each of the three small triangles equals the length of the adjacent side. (b) For a given triangle $\triangle ABC$, determine all inner points $P$ such that the perimeter of each of the three small triangles equals the length of the adjacent side. (c) For which inner point does the sum of the areas of the three small triangles attain a minimum? [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 4)[/i]

2011 IMAC Arhimede, 4

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

Inscribed circle of triangle $ABC$ touches sides $BC$, $CA$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. Let $AA_{1}$, $BB_{1}$ and $CC_{1}$ be the altitudes of the triangle $ABC$ and $M$, $N$ and $P$ be the incenters of triangles $AB_{1}C_{1}$, $BC_{1}A_{1}$ and $CA_{1}B_{1}$, respectively. a) Prove that $M$, $N$ and $P$ are orthocentres of triangles $AYZ$, $BZX$ and $CXY$, respectively. b) Prove that common external tangents of these incircles, different from triangle sides, are concurent at orthocentre of triangle $XYZ$.

2009 Indonesia TST, 2

Tags: geometry , incenter , triangle inequality , geometric inequality , angle bisector

Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that \[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. \]

2012 Middle European Mathematical Olympiad, 6

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

Let $ ABCD $ be a convex quadrilateral with no pair of parallel sides, such that $ \angle ABC = \angle CDA $. Assume that the intersections of the pairs of neighbouring angle bisectors of $ ABCD $ form a convex quadrilateral $ EFGH $. Let $ K $ be the intersection of the diagonals of $ EFGH$. Prove that the lines $ AB $ and $ CD $ intersect on the circumcircle of the triangle $ BKD $.

2009 Croatia Team Selection Test, 3

Tags: geometry , incenter , circumcircle

A triangle $ ABC$ is given with $ \left|AB\right| > \left|AC\right|$. Line $ l$ tangents in a point $ A$ the circumcirle of $ ABC$. A circle centered in $ A$ with radius $ \left|AC\right|$ cuts $ AB$ in the point $ D$ and the line $ l$ in points $ E, F$ (such that $ C$ and $ E$ are in the same halfplane with respect to $ AB$). Prove that the line $ DE$ passes through the incenter of $ ABC$.

Ukraine Correspondence MO - geometry, 2010.11

Tags: geometry , orthocenter , incenter , angle

Let $ABC$ be an acute-angled triangle in which $\angle BAC = 60^o$ and $AB> AC$. Let $H$ and $I$ denote the points of intersection of the altitudes and angle bisectors of this triangle, respectively. Find the ratio $\angle ABC: \angle AHI$.

Croatia MO (HMO) - geometry, 2017.3

Tags: geometry , incenter , arc midpoint , equal angles

In triangle $ABC$, $|AB| <|BC|$ holds. Point $I$ is the center of the circle inscribed in that triangle. Let $M$ be the midpoint of the side $AC$, and $N$ be the midpoint of the arc $AC$ of the circumcircle of that triangle containing point $B$. Prove that $\angle IMA = \angle INB$.

2008 AMC 12/AHSME, 20

Tags: geometry , ratio , trigonometry , incenter , inradius , analytic geometry , angle bisector

Triangle $ ABC$ has $ AC\equal{}3$, $ BC\equal{}4$, and $ AB\equal{}5$. Point $ D$ is on $ \overline{AB}$, and $ \overline{CD}$ bisects the right angle. The inscribed circles of $ \triangle ADC$ and $ \triangle BCD$ have radii $ r_a$ and $ r_b$, respectively. What is $ r_a/r_b$? $ \textbf{(A)}\ \frac{1}{28}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(B)}\ \frac{3}{56}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(C)}\ \frac{1}{14}\left(10\minus{}\sqrt{2}\right) \qquad \textbf{(D)}\ \frac{5}{56}\left(10\minus{}\sqrt{2}\right) \\ \textbf{(E)}\ \frac{3}{28}\left(10\minus{}\sqrt{2}\right)$

1987 IMO Longlists, 52

Tags: circumcircle , incenter , collinearity , geometry

Given a nonequilateral triangle $ABC$, the vertices listed counterclockwise, find the locus of the centroids of the equilateral triangles $A'B'C'$ (the vertices listed counterclockwise) for which the triples of points $A,B', C'; A',B, C';$ and $A',B', C$ are collinear. [i]Proposed by Poland.[/i]

2008 Iran Team Selection Test, 9

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

$ I_a$ is the excenter of the triangle $ ABC$ with respect to $ A$, and $ AI_a$ intersects the circumcircle of $ ABC$ at $ T$. Let $ X$ be a point on $ TI_a$ such that $ XI_a^2\equal{}XA.XT$. Draw a perpendicular line from $ X$ to $ BC$ so that it intersects $ BC$ in $ A'$. Define $ B'$ and $ C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

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.

2021 Mediterranean Mathematics Olympiad, 3

Tags: geometry , incenter , equilateral , fixed point , fixed

Let $ABC$ be an equiangular triangle with circumcircle $\omega$. Let point $F\in AB$ and point $E\in AC$ so that $\angle ABE+\angle ACF=60^{\circ}$. The circumcircle of triangle $AFE$ intersects the circle $\omega$ in the point $D$. The halflines $DE$ and $DF$ intersect the line through $B$ and $C$ in the points $X$ and $Y$. Prove that the incenter of the triangle $DXY$ is independent of the choice of $E$ and $F$. (The angles in the problem statement are not directed. It is assumed that $E$ and $F$ are chosen in such a way that the halflines $DE$ and $DF$ indeed intersect the line through $B$ and $C$.)

2014 China National Olympiad, 1

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

Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.

2023 Sharygin Geometry Olympiad, 4

Tags: geometry , incenter

Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$. Let $D'$ be the reflection of $D$ about $AC$. Prove that the line $D'E$ passes through the incenter of $ABC$.

2019 Taiwan TST Round 2, 6

Tags: geometry , incenter

Given a triangle $ \triangle{ABC} $ whose incenter is $ I $ and $ A $-excenter is $ J $. $ A' $ is point so that $ AA' $ is a diameter of $ \odot\left(\triangle{ABC}\right) $. Define $ H_{1}, H_{2} $ to be the orthocenters of $ \triangle{BIA'} $ and $ \triangle{CJA'} $. Show that $ H_{1}H_{2} \parallel BC $

2015 Tuymaada Olympiad, 7

Tags: geometry , incenter

$CL$ is bisector of $\angle C$ of $ABC$ and intersect circumcircle at $K$. $I$ - incenter of $ABC$. $IL=LK$. Prove, that $CI=IK$ [i]D. Shiryaev [/i]

2007 Junior Balkan MO, 2

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

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 $\angle{APD}$.

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 $OP\perp AP$

2013 Iran MO (3rd Round), 2

Tags: circumcircle , incenter , geometric transformation , geometry

Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.