Olympiad problems

2019 Mediterranean Mathematics Olympiad, 1

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 \[ \frac{1}{r_B} +\frac{1}{r_C} ~=~ 2\cdot\left( \frac1r +\frac1b +\frac1c\right)\]

1988 IMO Longlists, 70

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

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

2004 AMC 10, 22

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

A triangle with sides of $ 5$, $ 12$, and $ 13$ has both an inscibed and a circumscribed circle. What is the distance between the centers of those circles? $ \textbf{(A)}\ \frac{3\sqrt{5}}{2}\qquad \textbf{(B)}\ \frac{7}{2}\qquad \textbf{(C)}\ \sqrt{15}\qquad \textbf{(D)}\ \frac{\sqrt{65}}{2}\qquad \textbf{(E)}\ \frac{9}{2}$

2001 AIME Problems, 7

Tags: analytic geometry , geometry , inradius , similar triangles , incenter

Let $\triangle{PQR}$ be a right triangle with $PQ=90$, $PR=120$, and $QR=150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{10n}$. What is $n$?

2022 Bolivia Cono Sur TST, P4

Tags: pythagorean triple , inradius , number theory , geometry

Find all right triangles with integer sides and inradius 6.

2015 Sharygin Geometry Olympiad, P15

Tags: geometry , geometric inequalities , circumradius , inradius

The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.

2020 Malaysia IMONST 2, 1

Tags: right triangle , geometry , inradius

Prove that if $a$ and $b$ are legs, $c$ is the hypotenuse of a right triangle, then the radius of a circle inscribed in this triangle can be found by the formula $r = \frac12 (a + b - c)$.

2009 IMO Shortlist, 8

Tags: geometry , circumcircle , trigonometry , inradius , incenter

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]

1995 Romania Team Selection Test, 3

Tags: geometry , sidelenghts , altitude , inradius

The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.

2002 National Olympiad First Round, 13

Tags: geometry , trapezoid , inradius , polynomial , incenter , ratio , algebra

Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $|BC|+|AD| = 7$, $|AB| = 9$ and $|BC| = 14$. What is the ratio of the area of the triangle formed by $CD$, angle bisector of $\widehat{BCD}$ and angle bisector of $\widehat{CDA}$ over the area of the trapezoid? $ \textbf{a)}\ \dfrac{9}{14} \qquad\textbf{b)}\ \dfrac{5}{7} \qquad\textbf{c)}\ \sqrt 2 \qquad\textbf{d)}\ \dfrac{49}{69} \qquad\textbf{e)}\ \dfrac 13 $

2019 Jozsef Wildt International Math Competition, W. 68

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

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 $t\in [0,1]$

2011 NIMO Summer Contest, 8

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

Triangle $ABC$ with $\measuredangle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\tfrac{BE}{EF} = \tfrac{2}{3}$, then $\tfrac{CF}{FE} = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Lewis Chen [/i]

2014 India Regional Mathematical Olympiad, 4

Tags: inradius , inequalities , geometry

let $ABC$ be a right angled triangle with inradius $1$ find the minimum area of triangle $ABC$

1950 AMC 12/AHSME, 35

Tags: inradius , geometry

In triangle $ABC$, $AC=24$ inches, $BC=10$ inches, $AB=26$ inches. The radius of the inscribed circle is: $\textbf{(A)}\ 26\text{ in} \qquad \textbf{(B)}\ 4\text{ in} \qquad \textbf{(C)}\ 13\text{ in} \qquad \textbf{(D)}\ 8\text{ in} \qquad \textbf{(E)}\ \text{None of these}$

2009 Sharygin Geometry Olympiad, 18

Tags: incenter , rectangle , geometry , inradius

Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).

1989 Tournament Of Towns, (204) 2

Tags: geometry , incircle , median , inradius

In the triangle $ABC$ the median $AM$ is drawn. Is it possible that the radius of the circle inscribed in $\vartriangle ABM$ could be twice as large as the radius of the circle inscribed in $\vartriangle ACM$ ? ( D . Fomin , Leningrad)

2010 Korea National Olympiad, 3

Tags: trigonometry , power of a point , circumcircle , inradius , angle bisector , incenter , geometry

Let $ I $ be the incenter of triangle $ ABC $. The incircle touches $ BC, CA, AB$ at points $ P, Q, R $. A circle passing through $ B , C $ is tangent to the circle $I$ at point $ X $, a circle passing through $ C , A $ is tangent to the circle $I$ at point $ Y $, and a circle passing through $ A , B $ is tangent to the circle $I$ at point $ Z $, respectively. Prove that three lines $ PX, QY, RZ $ are concurrent.

2006 Dutch Mathematical Olympiad, 4

Tags: incircle , geometry , inradius

Given is triangle $ABC$ with an inscribed circle with center $M$ and radius $r$. The tangent to this circle parallel to $BC$ intersects $AC$ in $D$ and $AB$ in $E$. The tangent to this circle parallel to $AC$ intersects $AB$ in $F$ and $BC$ in $G$. The tangent to this circle parallel to $AB$ intersects $BC$ in $H$ and $AC$ in $K$. Name the centers of the inscribed circles of triangle $AED$, triangle $FBG$ and triangle $KHC$ successively $M_A, M_B, M_C$ and the rays successively $r_A, r_B$ and $r_C$. Prove that $r_A + r_B + r_C = r$.

Kyiv City MO Juniors 2003+ geometry, 2003.9.4

Tags: rhombus , parallelogram , geometry , 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)

2023 Yasinsky Geometry Olympiad, 4

Tags: angle bisector , inradius , geometry

The circle inscribed in triangle $ABC$ touches $AC$ at point $F$. The perpendicular from point $F$ on $BC$ intersects the bisector of angle $C$ at point $N$. Prove that segment $FN$ is equal to the radius of the circle inscribed in triangle $ABC$. (Oleksii Karliuchenko)

2020 Durer Math Competition Finals, 13

Tags: inscribed , inradius , geometry , square

In triangle $ABC$ we inscribe a square such that one of the sides of the square lies on the side $AC$, and the other two vertices lie on sides $AB$ and $BC$. Furthermore we know that $AC = 5$, $BC = 4$ and $AB = 3$. This square cuts out three smaller triangles from $\vartriangle ABC$. Express the sum of reciprocals of the inradii of these three small triangles as a fraction $p/q$ in lowest terms (i.e. with $p$ and $q$ coprime). What is $p + q$?

1978 IMO Longlists, 41

Tags: circumcircle , triangle , geometry , inradius , incenter

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 $ABC.$

2012 German National Olympiad, 5

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

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

Kvant 2021, M2648

Tags: inradius , geometry

The triangle $ABC$ is given. Consider the point $C'{}$ on the side $AB$ such that the segment $CC'$ divides the triangle into two triangles with equal radii of inscribed circles. Denote by $t_c$ the length of the segment $CC'$. Similarly, we define $t_a$ and $t_b$. Express the area of triangle $ABC$ in terms of $t_a,t_b$ and $t_c$. [i]Proposed by K. Mosevich[/i]

2012 All-Russian Olympiad, 2

Tags: perpendicular bisector , circumcircle , reflection , geometry , geometric transformation , inradius

The inscribed circle $\omega$ of the non-isosceles acute-angled triangle $ABC$ touches the side $BC$ at the point $D$. Suppose that $I$ and $O$ are the centres of inscribed circle and circumcircle of triangle $ABC$ respectively. The circumcircle of triangle $ADI$ intersects $AO$ at the points $A$ and $E$. Prove that $AE$ is equal to the radius $r$ of $\omega$.