Olympiad problems

2013 ELMO Shortlist, 8

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]

2024 Thailand TST, 2

Tags: geometry , nithi , incenter , mmp

Let $ABC$ be triangle with incenter $I$ . Let $AI$ intersect $BC$ at $D$. Point $P,Q$ lies inside triangle $ABC$ such that $\angle BPA + \angle CQA = 180^\circ$ and $B,Q,I,P,C$ concyclic in order . $BP$ intersect $CQ$ at $X$. Prove that the intersection of $(ABC)$ and $(APQ)$ lies on line $XD$.

2006 National Olympiad First Round, 25

Tags: geometry , incenter , angle bisector

Let $E$ be the midpoint of the side $[BC]$ of $\triangle ABC$ with $|AB|=7$, $|BC|=6$, and $|AC|=5$. The line, which passes through $E$ and is perpendicular to the angle bisector of $\angle A$, intersects $AB$ at $D$. What is $|AD|$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ \frac 92 \qquad\textbf{(D)}\ 3\sqrt 2 \qquad\textbf{(E)}\ \text{None of above} $

2020 Kazakhstan National Olympiad, 4

Tags: geometry , incenter , arc midpoint , incircle

The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.

1993 Balkan MO, 3

Tags: incenter , circumcircle , geometry

Circles $\mathcal C_1$ and $\mathcal C_2$ with centers $O_1$ and $O_2$, respectively, are externally tangent at point $\lambda$. A circle $\mathcal C$ with center $O$ touches $\mathcal C_1$ at $A$ and $\mathcal C_2$ at $B$ so that the centers $O_1$, $O_2$ lie inside $C$. The common tangent to $\mathcal C_1$ and $\mathcal C_2$ at $\lambda$ intersects the circle $\mathcal C$ at $K$ and $L$. If $D$ is the midpoint of the segment $KL$, show that $\angle O_1OO_2 = \angle ADB$. [i]Greece[/i]

2003 Gheorghe Vranceanu, 4

Tags: geometry , incenter , geometric inequality , inequalities

Let $ I $ be the incentre of $ ABC $ and $ D,E,F $ be the feet of the perpendiculars from $ I $ to $ BC,CA,AB, $ respectively. Show that $$ \frac{AB}{DE} +\frac{BC}{EF} +\frac{CA}{FD}\ge 6. $$

2016 NIMO Problems, 3

Tags: geometry , incenter

Right triangle $ABC$ has hypotenuse $AB = 26$, and the inscribed circle of $ABC$ has radius $5$. The largest possible value of $BC$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are both positive integers. Find $100m + n$. [i]Proposed by Jason Xia[/i]

2021 China Team Selection Test, 1

Tags: geometry , incenter , angle bisector

A cyclic quadrilateral $ABCD$ has circumcircle $\Gamma$, and $AB+BC=AD+DC$. Let $E$ be the midpoint of arc $BCD$, and $F (\neq C)$ be the antipode of $A$ [i]wrt[/i] $\Gamma$. Let $I,J,K$ be the incenter of $\triangle ABC$, the $A$-excenter of $\triangle ABC$, the incenter of $\triangle BCD$, respectively. Suppose that a point $P$ satisfies $\triangle BIC \stackrel{+}{\sim} \triangle KPJ$. Prove that $EK$ and $PF$ intersect on $\Gamma.$

2003 Silk Road, 2

Tags: geometry , perimeter , circumcircle , incenter , euler , parallelogram , trapezoid

Let $s=\frac{AB+BC+AC}{2}$ be half-perimeter of triangle $ABC$. Let $L$ and $N$be a point's on ray's $AB$ and $CB$, for which $AL=CN=s$. Let $K$ is point, symmetric of point $B$ by circumcenter of $ABC$. Prove, that perpendicular from $K$ to $NL$ passes through incenter of $ABC$. Solution for problem [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

2015 Sharygin Geometry Olympiad, P16

Tags: geometry , circumcircle , incenter

The diagonals of a convex quadrilateral divide it into four triangles. Restore the quadrilateral by the circumcenters of two adjacent triangles and the incenters of two mutually opposite triangles

2008 Bulgarian Autumn Math Competition, Problem 12.2

Tags: geometry , incenter , excircle , collinear points

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 $\angle ABC=60^{\circ}$.

1984 Vietnam National Olympiad, 3

Tags: trigonometry , incenter , geometry

Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha, \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively. $(a)$ Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$ $(b)$ Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$ $(c)$ Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.

2003 Korea - Final Round, 1

Tags: inequalities , geometry , incenter , trigonometry , inradius , perimeter , cyclic quadrilateral

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2016 Iran MO (3rd Round), 2

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

Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively. Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively. Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$. Prove that $XYII_a$ is cyclic.

Ukrainian TYM Qualifying - geometry, 2014.22

Tags: orthocenter , geometry , incenter , symmetric

In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .

2003 France Team Selection Test, 1

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

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

2009 Iran Team Selection Test, 1

Tags: incenter , inradius , trigonometry , geometry

Let $ ABC$ be a triangle and $ A'$ , $ B'$ and $ C'$ lie on $ BC$ , $ CA$ and $ AB$ respectively such that the incenter of $ A'B'C'$ and $ ABC$ are coincide and the inradius of $ A'B'C'$ is half of inradius of $ ABC$ . Prove that $ ABC$ is equilateral .

2006 Kyiv Mathematical Festival, 4

Tags: circumcircle , incenter , trigonometry , trig identities , law of sines , geometry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2001 JBMO ShortLists, 10

Tags: ratio , incenter , geometry

A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$.

2021 Yasinsky Geometry Olympiad, 5

Tags: incenter , bisects segment , construction , ruler , geometry

Circle $\omega$ is inscribed in the $\vartriangle ABC$, with center $I$. Using only a ruler, divide segment $AI$ in half. (Grigory Filippovsky)

2012 Oral Moscow Geometry Olympiad, 1

Tags: midpoint , geometry , incenter

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?

1978 IMO, 1

Tags: geometry , inradius , circumcircle , incenter , triangle

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

2011 Saudi Arabia IMO TST, 2

Tags: geometry , incenter , right angle , circumcircle

Let $ABC$ be a triangle with $AB\ne AC$. Its incircle has center $I$ and touches the side $BC$ at point $D$. Line $AI$ intersects the circumcircle $\omega$ of triangle $ABC$ at $M$ and $DM$ intersects again $\omega$ at $P$. Prove that $\angle API= 90^o$.

2008 Sharygin Geometry Olympiad, 4

Tags: incenter , geometry

(A.Zaslavsky) Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.

2016 JBMO Shortlist, 3

Tags: geometry , trapezoid , incenter

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.