Olympiad problems

2007 China Northern MO, 4

Tags: geometry , inradius , inequalities , area of a triangle , heron's formula , triangle inequality , algebra

The inradius of triangle $ ABC$ is $ 1$ and the side lengths of $ ABC$ are all integers. Prove that triangle $ ABC$ is right-angled.

2004 France Team Selection Test, 2

Tags: inequalities , incenter , trigonometry , 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 $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2010 Germany Team Selection Test, 3

Tags: geometry , incenter , circumcircle , trigonometry , inradius

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]

2007 USAMO, 6

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

Let $ABC$ be an acute triangle with $\omega,S$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_{A}$ is tangent internally to $S$ at $A$ and tangent externally to $\omega$. Circle $S_{A}$ is tangent internally to $S$ at $A$ and tangent internally to $\omega$. Let $P_{A}$ and $Q_{A}$ denote the centers of $\omega_{A}$ and $S_{A}$, respectively. Define points $P_{B}, Q_{B}, P_{C}, Q_{C}$ analogously. Prove that \[8P_{A}Q_{A}\cdot P_{B}Q_{B}\cdot P_{C}Q_{C}\leq R^{3}\; , \] with equality if and only if triangle $ABC$ is equilateral.

2002 Bulgaria National Olympiad, 2

Tags: circumcircle , inradius , incenter , pythagorean theorem , geometry

Consider the orthogonal projections of the vertices $A$, $B$ and $C$ of triangle $ABC$ on external bisectors of $ \angle ACB$, $ \angle BAC$ and $ \angle ABC$, respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$, prove that $r^2+p^2=d^2$ [i]Proposed by Alexander Ivanov[/i]

1986 Vietnam National Olympiad, 2

Tags: geometry , inequalities , circumcircle , inradius , trigonometry , 3d geometry , pyramid

Let $ R$, $ r$ be respectively the circumradius and inradius of a regular $ 1986$-gonal pyramid. Prove that \[ \frac{R}{r}\ge 1\plus{}\frac{1}{\cos\frac{\pi}{1986}}\] and find the total area of the surface of the pyramid when the equality occurs.

2014 Harvard-MIT Mathematics Tournament, 5

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

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 $P$.

2019 Yasinsky Geometry Olympiad, p6

Tags: geometry , inradius , geometric inequality

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

2008 Sharygin Geometry Olympiad, 3

Tags: inequalities , inradius , trigonometry , geometry

(R.Pirkuliev) Prove the inequality \[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.

Kvant 2021, M2635

Tags: inequalities , inradius , geometry

In the triangle $ABC$, the lengths of the sides $BC, CA$ and $AB$ are $a,b$ and $c{}$ respectively. Several segments are drawn from the vertex $C{}$, which cut the triangle $ABC$ into several triangles. Find the smallest number $M{}$ for which, with each such cut, the sum of the radii of the circles inscribed in triangles does not exceed $M{}$. [i]Porposed by O. Titov[/i]

1983 AMC 12/AHSME, 24

Tags: geometry , perimeter , trigonometry , inradius , sfft

How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal? $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ \text{infinitely many}$

1981 Vietnam National Olympiad, 1

Tags: trigonometry , perimeter , circumcircle , inradius , geometry

Prove that a triangle $ABC$ is right-angled if and only if \[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]

1970 AMC 12/AHSME, 27

Tags: geometry , inradius

In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }6$

Kyiv City MO Juniors 2003+ geometry, 2003.9.4

Tags: geometry , parallelogram , rhombus , inradius

The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than: a) parallelogram, b) rhombus? (Sharygin Igor)

2013 Turkey Team Selection Test, 3

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

Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that \[\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.\]

1998 Austrian-Polish Competition, 9

Tags: geometry , inradius , circumcircle , arc midpoint , midpoint

Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.

1978 IMO Longlists, 41

Tags: geometry , inradius , circumcircle , incenter , triangle

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

1995 Korea National Olympiad, Problem 3

Tags: inradius , equilateral triangle , maximum , geometry

Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.

2004 Iran MO (3rd Round), 29

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

Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.

2011 Albania Team Selection Test, 2

Tags: geometry , perimeter , calculus , integration , inradius , trigonometry , algorithm

The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.

2014 India PRMO, 12

Tags: geometry , inradius , sum , right angle

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ?

Estonia Open Senior - geometry, 2009.1.5

Tags: geometry , geometric inequality , 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 $r_1 <r_2 + r_3$.

2008 Kazakhstan National Olympiad, 2

Tags: circumcircle , inradius , trigonometry , incenter , geometry

Let $ \triangle ABC$ be a triangle and let $ K$ be some point on the side $ AB$, so that the tangent line from $ K$ to the incircle of $ \triangle ABC$ intersects the ray $ AC$ at $ L$. Assume that $ \omega$ is tangent to sides $ AB$ and $ AC$, and to the circumcircle of $ \triangle AKL$. Prove that $ \omega$ is tangent to the circumcircle of $ \triangle ABC$ as well.

2005 Germany Team Selection Test, 3

Tags: geometry , incenter , circumcircle , inequalities , inradius , euler , vector

Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality \[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\] and determine when equality holds.

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.