Olympiad problems

2023 Belarusian National Olympiad, 11.7

Tags: incenter , geometry

Let $\omega$ be the incircle of triangle $ABC$. Line $l_b$ is parallel to side $AC$ and tangent to $\omega$. Line $l_c$ is parallel to side $AB$ and tangent to $\omega$. It turned out that the intersection point of $l_b$ and $l_c$ lies on circumcircle of $ABC$ Find all possible values of $\frac{AB+AC}{BC}$

2004 France Team Selection Test, 2

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

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 $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2001 Iran MO (2nd round), 2

Tags: perpendicular bisector , circumcircle , incenter , geometry , trigonometry

In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?

2012 USA TSTST, 4

Tags: incenter , circumcircle , cyclic quadrilateral , geometry , complex number

In scalene triangle $ABC$, let the feet of the perpendiculars from $A$ to $BC$, $B$ to $CA$, $C$ to $AB$ be $A_1, B_1, C_1$, respectively. Denote by $A_2$ the intersection of lines $BC$ and $B_1C_1$. Define $B_2$ and $C_2$ analogously. Let $D, E, F$ be the respective midpoints of sides $BC, CA, AB$. Show that the perpendiculars from $D$ to $AA_2$, $E$ to $BB_2$ and $F$ to $CC_2$ are concurrent.

2004 Switzerland Team Selection Test, 3

Tags: circumcircle , geometry , incenter , triangle

Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$. [i]Proposed by Hojoo Lee, Korea[/i]

2005 Balkan MO, 1

Tags: geometry , incenter

Let $ABC$ be an acute-angled triangle whose inscribed circle touches $AB$ and $AC$ at $D$ and $E$ respectively. Let $X$ and $Y$ be the points of intersection of the bisectors of the angles $\angle ACB$ and $\angle ABC$ with the line $DE$ and let $Z$ be the midpoint of $BC$. Prove that the triangle $XYZ$ is equilateral if and only if $\angle A = 60^\circ$.

2005 District Olympiad, 4

Tags: incenter , angle bisector , geometry , circumcircle , quadratic , perpendicular bisector , ratio

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

2024 Oral Moscow Geometry Olympiad, 2

Tags: geometry , incenter

The bisector $BL$ was drawn in the triangle $ABC$. Let the points $I_1$ and $I_2$ be centers of the circles inscribed in the triangles $ABL$ and $CBL$, and the points $J_1$ and $J_2$ be centers of the excircles of these triangles touching the side $BL$. Prove that the points $I_1$, $I_2$, $J_1$ and $J_2$ lie on the same circle.

2011 Iran MO (3rd Round), 5

Tags: circumcircle , angle bisector , incenter , geometric transformation , geometry

Given triangle $ABC$, $D$ is the foot of the external angle bisector of $A$, $I$ its incenter and $I_a$ its $A$-excenter. Perpendicular from $I$ to $DI_a$ intersects the circumcircle of triangle in $A'$. Define $B'$ and $C'$ similarly. Prove that $AA',BB'$ and $CC'$ are concurrent. [i]proposed by Amirhossein Zabeti[/i]

2023 Peru MO (ONEM), 4

Tags: geometry , incenter

Let $ABC$ be an acute scalene triangle and $K$ be a point inside it that belongs to the bisector of the angle $\angle ABC$. Let$ P$ be the point where the line $AK$ intersects the line perpendicular to $AB$ that passes through $B$, and let $Q$ be the point where the line $CK$ intersects the line perpendicular to $CB$ that passes through $B$. Let $L$ be the foot of the perpendicular drawn from $K$ on the line $AC$. Prove that if $P Q$ is perpendicular to $BL$, then $K$ is the incenter of $ABC$.

2002 Moldova Team Selection Test, 3

Tags: geometry , incenter , similar triangles , arc

A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.

2008 Iran Team Selection Test, 9

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

$ 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.

2016 Junior Balkan MO, 1

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 $MN$.

2008 Harvard-MIT Mathematics Tournament, 9

Tags: ratio , geometry , incenter , angle bisector

Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.