Olympiad problems

2010 Germany Team Selection Test, 3

Tags: incenter , circumcircle , trigonometry , inradius , geometry

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]

1989 IMO Longlists, 28

Tags: inradius , geometry

In a triangle $ ABC$ for which $ 6(a\plus{}b\plus{}c)r^2 \equal{} abc$ holds and where $ r$ denotes the inradius of $ ABC,$ we consider a point M on the inscribed circle and the projections $ D,E, F$ of $ M$ on the sides $ BC\equal{}a, AC\equal{}b,$ and $ AB\equal{}c$ respectively. Let $ S, S_1$ denote the areas of the triangles $ ABC$ and $ DEF$ respectively. Find the maximum and minimum values of the quotient $ \frac{S}{S_1}$

2010 Mediterranean Mathematics Olympiad, 3

Tags: circumcircle , inradius , geometric transformation , reflection , incenter , euler , geometry

Let $A'\in(BC),$ $B'\in(CA),C'\in(AB)$ be the points of tangency of the excribed circles of triangle $\triangle ABC$ with the sides of $\triangle ABC.$ Let $R'$ be the circumradius of triangle $\triangle A'B'C'.$ Show that \[ R'=\frac{1}{2r}\sqrt{2R\left(2R-h_{a}\right)\left(2R-h_{b}\right)\left(2R-h_{c}\right)}\] where as usual, $R$ is the circumradius of $\triangle ABC,$ r is the inradius of $\triangle ABC,$ and $h_{a},h_{b},h_{c}$ are the lengths of altitudes of $\triangle ABC.$

2017 Harvard-MIT Mathematics Tournament, 8

Tags: geometry , circumcircle , inradius

Let $ABC$ be a triangle with circumradius $R=17$ and inradius $r=7$. Find the maximum possible value of $\sin \frac{A}{2}$.

2012 German National Olympiad, 5

Tags: geometry , inradius , 3d geometry , tetrahedron , inequalities , geometric inequality

Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]

2011 Postal Coaching, 1

Tags: geometry , incenter , geometric transformation , circumcircle , reflection , inradius

Let $I$ be the incentre of a triangle $ABC$ and $\Gamma_a$ be the excircle opposite $A$ touching $BC$ at $D$. If $ID$ meets $\Gamma_a$ again at $S$, prove that $DS$ bisects $\angle BSC$.

2016 India IMO Training Camp, 1

Tags: inradius , circumcircle , geometric inequality , geometry

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $A_1,B_1$ and $C_1$ be respectively the midpoints of the arcs $BAC,CBA$ and $ACB$ of $\Gamma$. Show that the inradius of triangle $A_1B_1C_1$ is not less than the inradius of triangle $ABC$.

1983 Bundeswettbewerb Mathematik, 2

Tags: geometry , inradius , circumradius , right triangle , construction

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

1989 IMO, 2

Tags: geometry , circumcircle , inradius , area , geometric inequality

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.

2004 Bulgaria Team Selection Test, 3

Tags: inradius , geometry

Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.

2003 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: inequalities , inradius , geometry

Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.

2018 Israel National Olympiad, 6

Tags: circles , inradius , geometry , tangent circle

In the corners of triangle $ABC$ there are three circles with the same radius. Each of them is tangent to two of the triangle's sides. The vertices of triangle $MNK$ lie on different sides of triangle $ABC$, and each edge of $MNK$ is also tangent to one of the three circles. Likewise, the vertices of triangle $PQR$ lie on different sides of triangle $ABC$, and each edge of $PQR$ is also tangent to one of the three circles (see picture below). Prove that triangles $MNK,PQR$ have the same inradius. [img]https://i.imgur.com/bYuBabS.png[/img]

Kyiv City MO 1984-93 - geometry, 1987.7.1

Tags: geometry , geometric inequality , inradius

The circle inscribed in the triangle $ABC$ touches the side BC at point $K$. Prove that the segment $AK$ is longer than the diameter of the circle.

2013 ELMO Shortlist, 1

Tags: incenter , circumcircle , inradius , geometry

Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral. [i]Proposed by Owen Goff[/i]

2024 Sharygin Geometry Olympiad, 15

Tags: algebra , geometry , ratio , circumcircle , inradius

The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

2019 Yasinsky Geometry Olympiad, p5

Tags: inequalities , geometry , inradius , geometric inequality , radius

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2 <r_a$ . (here $r$ is the radius of the circle inscribed in the triangle $ABC$, $r_a$ is the radius of an exscribed circle that touches the sides of $BC$). (Mykola Moroz)

1994 Baltic Way, 16

Tags: geometry , circumcircle , inradius , search , combinatorics

The Wonder Island is inhabited by Hedgehogs. Each Hedgehog consists of three segments of unit length having a common endpoint, with all three angles between them $120^{\circ}$. Given that all Hedgehogs are lying flat on the island and no two of them touch each other, prove that there is a finite number of Hedgehogs on Wonder Island.

VII Soros Olympiad 2000 - 01, 10.4

Tags: geometry , inradius , centroid

An acute-angled triangle is inscribed in a circle of radius $R$. The distance between the center of the circle and the point of intersection of the medians of the triangle is $d$. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle.

2005 Turkey MO (2nd round), 5

Tags: inequalities , inradius , perimeter , geometry

If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that \[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} \]

2012 JBMO TST - Turkey, 1

Tags: inequalities , inradius , cauchy inequality , geometry

Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that \[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]

1994 IMO Shortlist, 3

Tags: geometry , circumcircle , inradius , symmetry , incenter

A circle $ C$ has two parallel tangents $ L'$ and$ L"$. A circle $ C'$ touches $ L'$ at $ A$ and $ C$ at $ X$. A circle $ C"$ touches $ L"$ at $ B$, $ C$ at $ Y$ and $ C'$ at $ Z$. The lines $ AY$ and $ BX$ meet at $ Q$. Show that $ Q$ is the circumcenter of $ XYZ$

1988 IMO Longlists, 70

Tags: trigonometry , inradius , circumcircle , function , geometry , inequalities

$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]

2010 USAMO, 4

Tags: trigonometry , geometry , incenter , inradius

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

1993 IMO Shortlist, 4

Tags: inradius , incenter , trigonometry , law of sines , geometry

Given a triangle $ABC$, let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that \[\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}. \]

2001 Estonia National Olympiad, 3

Tags: incircle , inradius , ratio , geometry

A circle with center $I$ and radius $r$ is inscribed in a triangle $ABC$ with a right angle at $C$. Rays $AI$ and $CI$ meet the opposite sides at $D$ and $E$ respectively. Prove that $\frac{1}{AE}+\frac{1}{BD}=\frac{1}{r}$