Found problems: 1389
2010 Germany Team Selection Test, 2
Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too.
[i]Proposed by Mirsaleh Bahavarnia, Iran[/i]
2004 Iran MO (3rd Round), 29
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.
1993 Vietnam Team Selection Test, 1
Let $H$, $I$, $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$. When does the equality hold ?
2013 Korea - Final Round, 1
For a triangle $ \triangle ABC (\angle B > \angle C) $, $ D $ is a point on $ AC $ satisfying $ \angle ABD = \angle C $. Let $ I $ be the incenter of $ \triangle ABC $, and circumcircle of $ \triangle CDI $ meets $ AI $ at $ E ( \ne I )$. The line passing $ E $ and parallel to $ AB $ meets the line $ BD $ at $ P $. Let $ J $ be the incenter of $ \triangle ABD $, and $ A' $ be the point such that $ AI = IA' $. Let $ Q $ be the intersection point of $ JP $ and $ A'C $. Prove that $ QJ = QA' $.
2017 Estonia Team Selection Test, 3
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 $BDE$.
2012 India IMO Training Camp, 1
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]
2010 Lithuania National Olympiad, 2
Let $I$ be the incenter of a triangle $ABC$. $D,E,F$ are the symmetric points of $I$ with respect to $BC,AC,AB$ respectively. Knowing that $D,E,F,B$ are concyclic,find all possible values of $\angle B$.
2010 Belarus Team Selection Test, 8.3
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
2014 All-Russian Olympiad, 4
Given a triangle $ABC$ with $AB>BC$, let $ \Omega $ be the circumcircle. Let $M$, $N$ lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. Let $K$ be the intersection of $MN$ and $AC$. Let $P$ be the incentre of the triangle $AMK$ and $Q$ be the $K$-excentre of the triangle $CNK$. If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$.
[i]M. Kungodjin[/i]
2006 Iran MO (3rd Round), 1
Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.
1994 Tournament Of Towns, (401) 3
Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon.
(V Proizvolov)
2006 India National Olympiad, 1
In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that
(1) $IO$ is perpendicular to $BI$;
(2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.
2016 Saudi Arabia BMO TST, 2
Let $I$ be the incenter of an acute triangle $ABC$. Assume that $K_1$ is the point such that $AK_1 \perp BC$ and the circle with center $K_1$ of radius $K_1A$ is internally tangent to the incircle of triangle $ABC$ at $A_1$. The points $B_1, C_1$ are defined similarly.
a) Prove that $AA_1, BB_1, CC_1$ are concurrent at a point $P$.
b) Let $\omega_1,\omega_2,\omega_3$ be the excircles of triangle $ABC$ with respect to $A, B, C$, respectively. The circles $\gamma_1,\gamma_2\gamma_3$ are the reflections of $\omega_1,\omega_2,\omega_3$ with respect to the midpoints of $BC, CA, AB$, respectively. Prove that P is the radical center of $\gamma_1,\gamma_2,\gamma_3$.
1999 IMO Shortlist, 8
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
1988 Mexico National Olympiad, 6
Consider two fixed points $B,C$ on a circle $w$. Find the locus of the incenters of all triangles $ABC$ when point $A$ describes $w$.
2012 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$
respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$
such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove
that $XY = (b + c-a)/2$ where $BC = a, CA = b $and $AB = c$.
2004 Flanders Math Olympiad, 1
[u][b]The author of this posting is : Peter VDD[/b][/u]
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most of us didn't really expect to get this, but here it goes (flanders mathematical olympiad 2004, today)
triangle with sides 501m, 668m, 835m
how many lines can be draws so that the line halves both area and circumference?
2019 Yasinsky Geometry Olympiad, p4
In the triangle $ABC$, the side $BC$ is equal to $a$. Point $F$ is midpoint of $AB$, $I$ is the point of intersection of the bisectors of triangle $ABC$. It turned out that $\angle AIF = \angle ACB$. Find the perimeter of the triangle $ABC$.
(Grigory Filippovsky)
2010 Contests, 2
Let $ I$ be the incentre and $ O$ the circumcentre of a given acute triangle $ ABC$. The incircle is tangent to $ BC$ at $ D$. Assume that $ \angle B < \angle C$ and the segments $ AO$ and $ HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$. Let the intersection of the line $ OD$ and $ AH$ be $ E$. If the midpoint of $ CI$ is $ F$, prove that $ E,F,I,O$ are concyclic.
2009 All-Russian Olympiad, 2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
2006 India IMO Training Camp, 2
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.
[i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]
2014 Oral Moscow Geometry Olympiad, 6
A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.
2007 Romania Team Selection Test, 1
In a circle with center $O$ is inscribed a polygon, which is triangulated. Show that the sum of the squares of the distances from $O$ to the incenters of the formed triangles is independent of the triangulation.
2008 Iran MO (3rd Round), 1
Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' \equal{} BB' \equal{} CC' \equal{} x$. Prove that:
a) If $ ABC\sim A'B'C'$ then $ x \equal{} 2r$
b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x \equal{} R \plus{} d$ or $ x \equal{} R \minus{} d$.
(In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d \equal{} OI$)
1999 Slovenia National Olympiad, Problem 3
The incircle of a right triangle $ABC$ touches the hypotenuse $AB$ at a point $D$. Show that the area of $\triangle ABC$ equals $AD\cdot DB$.