Olympiad problems

1976 IMO Shortlist, 1

Let $ABC$ be a triangle with bisectors $AA_1,BB_1, CC_1$ ($A_1 \in BC$, etc.) and $M$ their common point. Consider the triangles $MB_1A, MC_1A,MC_1B,MA_1B,MA_1C,MB_1C$, and their inscribed circles. Prove that if four of these six inscribed circles have equal radii, then $AB = BC = CA.$

2000 Hungary-Israel Binational, 3

Tags: geometry , orthocenter , incircle

Let ${ABC}$ be a non-equilateral triangle. The incircle is tangent to the sides ${BC,CA,AB}$ at ${A_1,B_1,C_1}$, respectively, and M is the orthocenter of triangle ${A_1B_1C_1}$. Prove that ${M}$ lies on the line through the incenter and circumcenter of ${\vartriangle ABC}$.

1995 Poland - Second Round, 5

Tags: geometry , 3d geometry , tetrahedron , concyclic , incircle

The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.

1965 All Russian Mathematical Olympiad, 062

Tags: geometry , maximum , incircle

What is the maximal possible length of the segment, being cut out by the sides of the triangle on the tangent to the inscribed circle, being drawn parallel to the base, if the triangle's perimeter equals $2p$?

Durer Math Competition CD Finals - geometry, 2015.C1

Tags: incircle , obtuse , tangent , geometry

Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?

2015 Belarus Team Selection Test, 3

Tags: geometry , incircle , concyclic

Let the incircle of the triangle $ABC$ touch the side $AB$ at point $Q$. The incircles of the triangles $QAC$ and $QBC$ touch $AQ,AC$ and $BQ,BC$ at points $P,T$ and $D,F$ respectively. Prove that $PDFT$ is a cyclic quadrilateral. I.Gorodnin

2000 Poland - Second Round, 4

Tags: geometry , incircle , incenter

Point $I$ is incenter of triangle $ABC$ in which $AB \neq AC$. Lines $BI$ and $CI$ intersect sides $AC$ and $AB$ in points $D$ and $E$, respectively. Determine all measures of angle $BAC$, for which may be $DI = EI$.

2015 BMT Spring, 7

Tags: geometry , incircle , angle

In $ \vartriangle ABC$, $\angle B = 46^o$ and $\angle C = 48^o$ . A circle is inscribed in $ \vartriangle ABC$ and the points of tangency are connected to form $PQR$. What is the measure of the largest angle in $\vartriangle P QR$?

2006 Tournament of Towns, 1

Tags: heptagon , 17-gon , incircle , area , circumcircle , regular polygon , geometry

Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)

1999 Israel Grosman Mathematical Olympiad, 3

Tags: incircle , geometry , equilateral

For every triangle $ABC$, denote by $D(ABC)$ the triangle whose vertices are the tangency points of the incircle of $\vartriangle ABC$ with the sides. Assume that $\vartriangle ABC$ is not equilateral. (a) Prove that $D(ABC)$ is also not equilateral. (b) Find in the sequence $T_1 = \vartriangle ABC, T_{k+1} = D(T_k)$ for $k \in N$ a triangle whose largest angle $\alpha$ satisfies $0 < \alpha -60^o < 0.0001^o$

2014 ELMO Shortlist, 7

Tags: mixtilinear incircle , geometry , concurrency , incircle , mixtilinear circle

Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent. [i]Proposed by Robin Park[/i]

Kyiv City MO 1984-93 - geometry, 1989.9.1

Tags: geometry , area , incircle

The perimeter of the triangle $ABC$ is equal to $2p$, the length of the side$ AC$ is equal to $b$, the angle $ABC$ is equal to $\beta$. A circle with center at point $O$, inscribed in this triangle, touches the side $BC$ at point $K$. Calculate the area of the triangle $BOK$.

2011 Sharygin Geometry Olympiad, 6

Tags: incenter , altitude , orthocenter , contact triangle , incircle , geometry

Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ in points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.

1952 Moscow Mathematical Olympiad, 223

Tags: geometry , incircle , tangent circle

In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.

1982 IMO, 2

Tags: geometry , triangle , midpoint , incircle , concurrency

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

2010 Sharygin Geometry Olympiad, 4

Tags: geometry , construction , incircle

In triangle $ABC$, touching points $A', B'$ of the incircle with $BC, AC$ and common point $G$ of segments $AA'$ and $BB'$ were marked. After this the triangle was erased. Restore it by the ruler and the compass.

2018 Vietnam Team Selection Test, 6

Tags: geometry , incircle , excircle

Triangle $ABC$ circumscribed $(O)$ has $A$-excircle $(J)$ that touches $AB,\ BC,\ AC$ at $F,\ D,\ E$, resp. a. $L$ is the midpoint of $BC$. Circle with diameter $LJ$ cuts $DE,\ DF$ at $K,\ H$. Prove that $(BDK),\ (CDH)$ has an intersecting point on $(J)$. b. Let $EF\cap BC =\{G\}$ and $GJ$ cuts $AB,\ AC$ at $M,\ N$, resp. $P\in JB$ and $Q\in JC$ such that $$\angle PAB=\angle QAC=90{}^\circ .$$ $PM\cap QN=\{T\}$ and $S$ is the midpoint of the larger $BC$-arc of $(O)$. $(I)$ is the incircle of $ABC$. Prove that $SI\cap AT\in (O)$.

2013 Bosnia and Herzegovina Junior BMO TST, 3

Tags: geometry , incircle , angle bisector

Let $M$ and $N$ be touching points of incircle with sides $AB$ and $AC$ of triangle $ABC$, and $P$ intersection point of line $MN$ and angle bisector of $\angle ABC$. Prove that $\angle BPC =90 ^{\circ}$

2024 New Zealand MO, 6

Tags: geometry , incircle

Let $\omega$ be the incircle of scalene triangle $ABC$. Let $\omega$ be tangent to $AB$ and $AC$ at points $X$ and $Y$. Construct points $X^\prime$ and $Y^\prime$ on line segments $AB$ and $AC$ respectively such that $AX^\prime=XB$ and $AY^\prime=YC$. Let line $CX^\prime$ intersects $\omega$ at points $P,Q$ such that $P$ is closer to $C$ than $Q$. Also let $R^\prime$ be the intersection of lines $CX^\prime$ and $BY^\prime$. Prove that $CP=RX^\prime$.

2006 Abels Math Contest (Norwegian MO), 4

Tags: right triangle , mixtilinear circle , circumcircle , inradius , incircle , tangent circle

Let $\gamma$ be the circumscribed circle about a right-angled triangle $ABC$ with right angle $C$. Let $\delta$ be the circle tangent to the sides $AC$ and $BC$ and tangent to the circle $\gamma$ internally. (a) Find the radius $i$ of $\delta$ in terms of $a$ when $AC$ and $BC$ both have length $a$. (b) Show that the radius $i$ is twice the radius of the inscribed circle of $ABC$.

2016 Sharygin Geometry Olympiad, P14

Tags: geometry , symmetry , symmedian , circumcircle , incircle , concurrency , tangent circle

Let a triangle $ABC$ be given. Consider the circle touching its circumcircle at $A$ and touching externally its incircle at some point $A_1$. Points $B_1$ and $C_1$ are defined similarly. a) Prove that lines $AA_1, BB_1$ and $CC1$ concur. b) Let $A_2$ be the touching point of the incircle with $BC$. Prove that lines $AA_1$ and $AA_2$ are symmetric about the bisector of angle $\angle A$.

2011 Portugal MO, 5

Tags: geometry , incircle , projection , angle bisector

Let $[ABC]$ be a triangle, $D$ be the orthogonal projection of $B$ on the bisector of $\angle ACB$ and $E$ the orthogonal projection of $C$ on the bisector of $\angle ABC$ . Prove that $DE$ intersects the sides $[AB]$ and $[AC]$ at the touchpoints of the circle inscribed in the triangle $[ABC]$.

1982 IMO Shortlist, 13

Tags: triangle , midpoint , incircle , concurrency , geometry

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

1989 All Soviet Union Mathematical Olympiad, 489

Tags: geometry , incircle , common tangent , tangent

The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.

2016 Sharygin Geometry Olympiad, P9

Tags: incircle , right triangle , concurrency , geometry

Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively, $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.