Olympiad problems

2007 India National Olympiad, 6

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

If $ x$, $ y$, $ z$ are positive real numbers, prove that \[ \left(x \plus{} y \plus{} z\right)^2 \left(yz \plus{} zx \plus{} xy\right)^2 \leq 3\left(y^2 \plus{} yz \plus{} z^2\right)\left(z^2 \plus{} zx \plus{} x^2\right)\left(x^2 \plus{} xy \plus{} y^2\right) .\]

2004 Korea - Final Round, 2

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

An acute triangle $ABC$ has circumradius $R$, inradius $r$. $A$ is the biggest angle among $A,B,C$. Let $M$ be the midpoint of $BC$, and $X$ be the intersection of two lines that touches circumcircle of $ABC$ and goes through $B,C$ respectively. Prove the following inequality : $ \frac{r}{R} \geq \frac{AM}{AX}$.

Estonia Open Senior - geometry, 2009.1.5

Tags: geometric inequality , geometry , inradius

Let any point $D$ be chosen on the side $BC$ of the triangle $ABC$. Let the radii of the incircles of the triangles $ABC, ABD$ and $ACD$ be $r_1, r_2$ and $r_3$. Prove that $r_1 <r_2 + r_3$.

2000 Tournament Of Towns, 2

Tags: isosceles , geometry , inradius

In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$. (RK Gordin)

2018 Oral Moscow Geometry Olympiad, 1

Tags: right triangle , geometry , projection , inradius

In a right triangle $ABC$ with a right angle $C$, let $AK$ and $BN$ be the angle bisectors. Let $D,E$ be the projections of $C$ on $AK, BN$ respectively. Prove that the length of the segment $DE$ is equal to the radius of the inscribed circle.

1985 Polish MO Finals, 4

Tags: geometry , inequalities , inradius , geometric inequalities

$P$ is a point inside the triangle $ABC$ is a triangle. The distance of $P$ from the lines $BC, CA, AB$ is $d_a, d_b, d_c$ respectively. If $r$ is the inradius, show that $$\frac{2}{ \frac{1}{d_a} + \frac{1}{d_b} + \frac{1}{d_c}} < r < \frac{d_a + d_b + d_c}{2}$$

2008 Sharygin Geometry Olympiad, 6

Tags: geometry , circumcircle , inradius , trigonometry

(B.Frenkin) The product of two sides in a triangle is equal to $ 8Rr$, where $ R$ and $ r$ are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than $ 60^{\circ}$.

1988 IMO Longlists, 37

Tags: tetrahedron , geometry , incenter , inradius , 3d geometry , sphere , trigonometry

[b]i.)[/b] Four balls of radius 1 are mutually tangent, three resting on the floor and the fourth resting on the others. A tedrahedron, each of whose edges has length $ s,$ is circumscribed around the balls. Find the value of $ s.$ [b]ii.)[/b] Suppose that $ ABCD$ and $ EFGH$ are opposite faces of a retangular solid, with $ \angle DHC \equal{} 45^{\circ}$ and $ \angle FHB \equal{} 60^{\circ}.$ Find the cosine of $ \angle BHD.$

1991 Hungary-Israel Binational, 2

Tags: perimeter , geometry , perpendicular bisector , inradius

The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.

2012 Centers of Excellency of Suceava, 4

Tags: geometry , incenter , circumcircle , inradius

Let $ O $ be the circumcenter of a triangle $ ABC $ with $ \angle BAC=60^{\circ } $ whose incenter is denoted by $ I. $ Let $ B_1,C_1 $ be the intersection of $ BI,CI $ with the circumcircle of $ ABC, $ respectively. Denote by $ O_1,O_2 $ the circumcenters of $ BIC_1,CIB_1, $ respectively. Show that $ O_1,I,O,O_2 $ are collinear. [i]Cătălin Țigăeru[/i]

1993 Austrian-Polish Competition, 9

Tags: incircle , geometry , equilateral triangle , equilateral , inradius

Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.

2010 Germany Team Selection Test, 3

Tags: incenter , circumcircle , inradius , trigonometry , 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]

1992 Nordic, 3

Tags: geometry , inradius , minimum

Prove that among all triangles with inradius $1$, the equilateral one has the smallest perimeter .

2019 Jozsef Wildt International Math Competition, W. 60

Tags: inequalities , tetrahedron , inradius , exradius , 3d geometry

In all tetrahedron $ABCD$ holds [list=1] [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(h_a-r)^2}{(h_a^n-r^n)(h_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [*] $(n(n+2))^{\frac{1}{n}} \sum \limits_{cyc} \left(\frac{(r_a-r)^2}{(r_a^n-r^n)(r_a^{n+2}-r^{n+2})}\right)^{\frac{1}{n}}\leq \frac{1}{r^2}$ [/list] for all $n\in \mathbb{N}^*$

2008 ITest, 32

Tags: inradius , perimeter , floor function , geometry

A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.

2017 QEDMO 15th, 10

Tags: equal segments , geometry , inradius

Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .

2006 Abels Math Contest (Norwegian MO), 4

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

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 $ABC$.

1997 Turkey Team Selection Test, 1

Tags: incenter , inradius , area of a triangle , geometry

In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.

2004 Bulgaria Team Selection Test, 3

Tags: geometry , inradius

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.

2013 ELMO Shortlist, 1

Tags: inradius , geometry , incenter , circumcircle

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]

2014 Harvard-MIT Mathematics Tournament, 5

Tags: analytic geometry , 3d geometry , sphere , inradius , geometry

Let $\mathcal{C}$ be a circle in the $xy$ plane with radius $1$ and center $(0, 0, 0)$, and let $P$ be a point in space with coordinates $(3, 4, 8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.

1986 IMO Longlists, 70

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

Let $ABCD$ be a tetrahedron having each sum of opposite sides equal to $1$. Prove that \[r_A + r_B + r_C + r_D \leq \frac{\sqrt 3}{3}\] where $r_A, r_B, r_C, r_D$ are the inradii of the faces, equality holding only if $ABCD$ is regular.

2014 Brazil National Olympiad, 1

Tags: geometry , rhombus , trigonometry , inradius , incenter

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

2019 Jozsef Wildt International Math Competition, W. 59

Tags: tetrahedron , geometry , inequalities , 3d geometry , circumradius , inradius

In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).

2003 India IMO Training Camp, 8

Tags: geometry , inradius

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.