Olympiad problems

2010 Sharygin Geometry Olympiad, 18

Tags: parallelogram , incenter , geometric transformation , homothety , projective geometry , geometry

A point $B$ lies on a chord $AC$ of circle $\omega.$ Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F,$ and the rays $AD$ and $CE$ do in a point $G.$ Prove that the line $FG$ passes through the midpoint of the segment $AC.$

III Soros Olympiad 1996 - 97 (Russia), 9.6

Tags: geometry , incenter , orthocenter

In triangle $ABC$, angle $B$ is not right. The circle inscribed in $ABC$ touches $AB$ and $BC$ at points $C_1$ and $A_1$, and the feet of the altitudes drawn to the sides $AB$ and $BC$ are points $C_2$ and $A_2$. Prove that the intersection point of the altitudes of triangle $A_1BC_1$ is the center of the circle inscribed in triangle $A_2BC_2$.

2016 IMO Shortlist, G4

Tags: geometry , incenter , reflection , inversion , cyclic quadrilateral

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 $BDE$.

2017 Iran MO (3rd round), 1

Tags: circumcircle , incenter , geometry

Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.

2024 India Iran Friendly Math Competition, 2

Tags: incentre , geometry , incenter

Let $ABCD$ be a cyclic quadrilateral with circumcentre $O_1$. The diagonals $AC$ and $BD$ meet at point $P$. Suppose the four incentres of triangles $PAB, PBC, PCD, PDA$ lie on a circle with centre $O_2$. Prove that $P, O_1, O_2$ are collinear. [i]Proposed by Shantanu Nene[/i]

2025 China National Olympiad, 2

Tags: incenter , circumcircle , homothety , geometry

Let $ABC$ be a triangle with incenter $I$. Denote the midpoints of $AI$, $AC$ and $CI$ by $L$, $M$ and $N$ respectively. Point $D$ lies on segment $AM$ such that $BC= BD$. Let the incircle of triangle $ABD$ be tangent to $AD$ and $BD$ at $E$ and $F$ respectively. Denote the circumcenter of triangle $AIC$ by $J$, and the circumcircle of triangle $JMD$ by $\omega$. Lines $MN$ and $JL$ meet $\omega$ again at $P$ and $Q$ respectively. Prove that $PQ$, $LN$ and $EF$ are concurrent.

2020 Dutch IMO TST, 1

Tags: geometry , incenter , equal angles

In acute-angled triangle $ABC, I$ is the center of the inscribed circle and holds $| AC | + | AI | = | BC |$. Prove that $\angle BAC = 2 \angle ABC$.

2024 Abelkonkurransen Finale, 4b

Tags: incenter , cyclic , pentagon , geometry

The pentagons $P_1P_2P_3P_4P_5$ and$I_1I_2I_3I_4I_5$ are cyclic, where $I_i$ is the incentre of the triangle $P_{i-1}P_iP_{i+1}$ (reckoned cyclically, that is $P_0=P_5$ and $P_6=P_1$). Show that the lines $P_1I_1, P_2I_2, P_3I_3, P_4I_4$ and $P_5I_5$ meet in a single point.

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 .

2024 TASIMO, 1

Tags: incenter , geometry

Let $ABC$ be a triangle with $AB<AC$ and incenter $I.$ A point $D$ lies on segment $AC$ such that $AB=AD,$ and the line $BI$ intersects $AC$ at $E.$ Suppose the line $CI$ intersects $BD$ at $F,$ and $G$ lies on segment $DI$ such that $FD=FG.$ Prove that the lines $AG$ and $EF$ intersect on the circumcircle of triangle $CEI.$ \\ Proposed by Avan Lim Zenn Ee, Malaysia

2024 Dutch IMO TST, 2

Tags: ratio , incenter , sides of a triangle , geometry

Let $ABC$ be a triangle. A point $P$ lies on the segment $BC$ such that the circle with diameter $BP$ passes through the incenter of $ABC$. Show that $\frac{BP}{PC}=\frac{c}{s-c}$ where $c$ is the length of segment $AB$ and $2s$ is the perimeter of $ABC$.

2019 Yasinsky Geometry Olympiad, p4

Tags: geometry , perimeter , equal angles , incenter

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)

2022 Yasinsky Geometry Olympiad, 6

Tags: geometry , incenter , fixed point , fixed , circumcircle

Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$. (Matvii Kurskyi)

2015 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , incenter

Let $ABC$ be a triangle with $AB \ne AC$ and $ I$ its incenter. Let $M$ be the midpoint of the side $BC$ and $D$ the projection of $I$ on $BC.$ The line $AI$ intersects the circle with center $M$ and radius $MD$ at $P$ and $Q.$ Prove that $\angle BAC + \angle PMQ = 180^{\circ}.$