Olympiad problems

2008 Mexico National Olympiad, 2

Consider a circle $\Gamma$, a point $A$ on its exterior, and the points of tangency $B$ and $C$ from $A$ to $\Gamma$. Let $P$ be a point on the segment $AB$, distinct from $A$ and $B$, and let $Q$ be the point on $AC$ such that $PQ$ is tangent to $\Gamma$. Points $R$ and $S$ are on lines $AB$ and $AC$, respectively, such that $PQ\parallel RS$ and $RS$ is tangent to $\Gamma$ as well. Prove that $[APQ]\cdot[ARS]$ does not depend on the placement of point $P$.

2007 IberoAmerican, 2

Tags: geometry , incenter , inradius , trigonometry , circumcircle , trig identities , law of sines

Let $ ABC$ be a triangle with incenter $ I$ and let $ \Gamma$ be a circle centered at $ I$, whose radius is greater than the inradius and does not pass through any vertex. Let $ X_{1}$ be the intersection point of $ \Gamma$ and line $ AB$, closer to $ B$; $ X_{2}$, $ X_{3}$ the points of intersection of $ \Gamma$ and line $ BC$, with $ X_{2}$ closer to $ B$; and let $ X_{4}$ be the point of intersection of $ \Gamma$ with line $ CA$ closer to $ C$. Let $ K$ be the intersection point of lines $ X_{1}X_{2}$ and $ X_{3}X_{4}$. Prove that $ AK$ bisects segment $ X_{2}X_{3}$.

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? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }6$

V Soros Olympiad 1998 - 99 (Russia), 10.5

Tags: geometry , inradius

The radius of the circle inscribed in triangle $ABC$ is equal to $r$. This circle is tangent to $BC$ at point $M$ and divides the segment $AM$ in ratio $k$ (starting from vertex $A$). Find the sum of the radii of the circles inscribed in triangles $AMB$ and $AMC$.

2005 Italy TST, 2

Tags: inradius , inequalities , rearrangement inequality , geometry

$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. $(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.

2014 Contests, 1

Tags: geometry , incenter , trigonometry , rhombus , inradius

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.

2008 Hong kong National Olympiad, 3

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

$ \Delta ABC$ is a triangle such that $ AB \neq AC$. The incircle of $ \Delta ABC$ touches $ BC, CA, AB$ at $ D, E, F$ respectively. $ H$ is a point on the segment $ EF$ such that $ DH \bot EF$. Suppose $ AH \bot BC$, prove that $ H$ is the orthocentre of $ \Delta ABC$. Remark: the original question has missed the condition $ AB \neq AC$

2019 Mediterranean Mathematics Olympiad, 1

Tags: geometry , incircle , inradius

Let $\Delta ABC$ be a triangle with angle $\angle CAB=60^{\circ}$, let $D$ be the intersection point of the angle bisector at $A$ and the side $BC$, and let $r_B,r_C,r$ be the respective radii of the incircles of $ABD$, $ADC$, $ABC$. Let $b$ and $c$ be the lengths of sides $AC$ and $AB$ of the triangle. Prove that \[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]

2004 Moldova Team Selection Test, 7

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

Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter. Let $P$ be a point on the segment $OH$. Prove that $6r\leq PA+PB+PC\leq 3R$, where $r$ is the inradius and $R$ the circumradius of triangle $ABC$. [b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)

1986 IMO Shortlist, 21

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

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.

2005 Moldova Team Selection Test, 1

Tags: geometry , circumcircle , inradius , inequalities , trigonometry , trig identities , law of sines

Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that $\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$, where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.

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 Online Math Open Problems, 32

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

In $\triangle ABC$ with incenter $I$, $AB = 61$, $AC = 51$, and $BC=71$. The circumcircles of triangles $AIB$ and $AIC$ meet line $BC$ at points $D$ ($D \neq B$) and $E$ ($E \neq C$), respectively. Determine the length of segment $DE$. [i]James Tao[/i]

2007 Pre-Preparation Course Examination, 1

Tags: inequalities , inradius , geometry

$D$ is an arbitrary point inside triangle $ABC$, and $E$ is inside triangle $BDC$. Prove that \[\frac{S_{DBC}}{(P_{DBC})^{2}}\geq\frac{S_{EBC}}{(P_{EBC})^{2}}\]

1993 Spain Mathematical Olympiad, 3

Tags: geometry , inradius , circumradius , geometric inequality

Prove that in every triangle the diameter of the incircle is not greater than the radius of the circumcircle.

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.

1927 Eotvos Mathematical Competition, 3

Tags: geometry , inradius , exradius , geometric inequalities

Consider the four circles tangent to all three lines containing the sides of a triangle $ABC$; let $k$ and $k_c$ be those tangent to side $AB$ between $A$ and $B$. Prove that the geometric mean of the radii of k and $k_c$, does not exceed half the length of $AB$.

2020 Jozsef Wildt International Math Competition, W9

Tags: triangle , inequalities , geometry , inradius

In any triangle $ABC$ prove that the following relationship holds: $$\begin{vmatrix}(b+c)^2&a^2&a^2\\b^2&(c+a)^2&b^2\\c^2&c^2&(a+b)^2\end{vmatrix}\ge93312r^6$$ [i]Proposed by D.M. Bătinețu-Giurgiu and Daniel Sitaru[/i]

2008 ISI B.Stat Entrance Exam, 5

Tags: geometry , inradius

Suppose $ABC$ is a triangle with inradius $r$. The incircle touches the sides $BC, CA,$ and $AB$ at $D,E$ and $F$ respectively. If $BD=x, CE=y$ and $AF=z$, then show that \[r^2=\frac{xyz}{x+y+z}\]

1988 IMO Longlists, 82

Tags: inradius , pythagorean theorem , geometry

The triangle $ABC$ has a right angle at $C.$ The point $P$ is located on segment $AC$ such that triangles $PBA$ and $PBC$ have congruent inscribed circles. Express the length $x = PC$ in terms of $a = BC, b = CA$ and $c = AB.$

2019 Jozsef Wildt International Math Competition, W. 68

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

In all tetrahedron $ABCD$ holds [list=1] [*] $\displaystyle{\sum \limits_{cyc}\frac{h_a-r}{h_a+r}\geq \sum \limits_{cyc}\frac{h_a^t-r^t}{(h_a+r)^t}}$ [*] $\displaystyle{\sum \limits_{cyc}\frac{2r_a-r}{2r_a+r}\geq \sum \limits_{cyc}\frac{2r_a^t-r^t}{(2r_a+r)^t}}$ [/list] for all $t\in [0,1]$

2006 Oral Moscow Geometry Olympiad, 4

Tags: geometry , equal circles , inradius , construction

An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal. (M. Volchkevich)

2016 India National Olympiad, P5

Tags: geometry , inradius , trigonometry , equation

Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that \[ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. \]

2006 Sharygin Geometry Olympiad, 8

Tags: geometry , square , segment , tangential quadrilateral , inradius

The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$. Find the length of $AB$.

Kyiv City MO Juniors Round2 2010+ geometry, 2019.9.31

Tags: geometry , inradius , incircle

A circle $k$ of radius $r$ is inscribed in $\vartriangle ABC$, tangent to the circle $k$, which are parallel respectively to the sides $AB, BC$ and $CA$ intersect the other sides of $\vartriangle ABC$ at points $M, N; P, Q$ and $L, T$ ($P, T \in AB$, $L, N \in BC$ and $M, Q\in AC$). Denote by $r_1,r_2,r_3$ the radii of inscribed circles in triangles $MNC, PQA$ and $LTB$. Prove that $r_1+r_2+r_3=r$.