Olympiad problems

1976 Vietnam National Olympiad, 2

Find all triangles $ABC$ such that $\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}$, where, as usual, $a, b, c$ are the lengths of sides $BC, CA, AB$ and $R$ is the circumradius.

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.

2014 Dutch Mathematical Olympiad, 5

Tags: geometry , rectangle , combinatorics , combinatorial geometry , circumradius

We consider the ways to divide a $1$ by $1$ square into rectangles (of which the sides are parallel to those of the square). All rectangles must have the same circumference, but not necessarily the same shape. a) Is it possible to divide the square into 20 rectangles, each having a circumference of $2:5$? b) Is it possible to divide the square into 30 rectangles, each having a circumference of $2$?

2005 Spain Mathematical Olympiad, 3

Tags: inradius , circumradius , sides of a triangle , geometry

In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that: a) $0^o \le A \le 60^o$. b) The height relative to side $a$ is three times the inradius $r$. c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.

2016 Oral Moscow Geometry Olympiad, 6

Tags: ratio , geometry , circumradius , circumcircle , symmetry

Given an acute triangle $ABC$. Let $A'$ be a point symmetric to $A$ with respect to $BC, O_A$ is the center of the circle passing through $A$ and the midpoints of the segments $A'B$ and $A'C. O_B$ and $O_C$ points are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles $ABC$ and $O_AO_BO_C$.

1959 AMC 12/AHSME, 43

Tags: circumcircle , area of a triangle , heron's formula , circumradius , circles , geometry

The sides of a triangle are $25,39,$ and $40$. The diameter of the circumscribed circle is: $ \textbf{(A)}\ \frac{133}{3}\qquad\textbf{(B)}\ \frac{125}{3}\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 41\qquad\textbf{(E)}\ 40 $

1988 Tournament Of Towns, (195) 2

Tags: geometry , circumcircle , circumradius , orthocenter

Let $N$ be the orthocentre of triangle $ABC$ (i .e. the point where the altitudes meet). Prove that the circumscribed circles of triangles $ABN, ACN$ and $BCN$ each have equal radius.

2009 Sharygin Geometry Olympiad, 2

Tags: geometry , cyclic quadrilateral , circumradius

A cyclic quadrilateral is divided into four quadrilaterals by two lines passing through its inner point. Three of these quadrilaterals are cyclic with equal circumradii. Prove that the fourth part also is cyclic quadrilateral and its circumradius is the same. (A.Blinkov)

1996 Israel National Olympiad, 5

Tags: equilateral , circumradius , inradius , geometry

Suppose that the circumradius $R$ and the inradius $r$ of a triangle $ABC$ satisfy $R = 2r$. Prove that the triangle is equilateral.

1985 All Soviet Union Mathematical Olympiad, 412

Tags: circumcircle , equal circles , circumradius , geometry

One of two circumferences of radius $R$ comes through $A$ and $B$ vertices of the $ABCD$ parallelogram. Another comes through $B$ and $D$. Let $M$ be another point of circumferences intersection. Prove that the circle circumscribed around $AMD$ triangle has radius $R$.

2018 India PRMO, 7

Tags: geometry , hexagon , distance , circumradius

A point $P$ in the interior of a regular hexagon is at distances $8,8,16$ units from three consecutive vertices of the hexagon, respectively. If $r$ is radius of the circumscribed circle of the hexagon, what is the integer closest to $r$?

2017 Argentina National Math Olympiad Level 2, 5

Tags: equal segments , angle bisector , circumradius , geometry

Let $ABCD$ be a convex quadrilateral with $AB = BD = 8$ and $CD = DA = 6$. Let $P$ be a point on side $AB$ such that $DP$ is bisector of angle $\angle ADB$ and let $Q$ be a point on side $BC$ such that $DQ$ is bisector of angle $\angle CDB$. Calculate the radius of the circumcircle of triangle $DPQ$. Note: The circumcircle of a triangle is the circle that passes through its three vertices.

1935 Moscow Mathematical Olympiad, 011

Tags: circumradius , geometry , radius , circumcircle

In $\vartriangle ABC$, two straight lines drawn from an arbitrary point $D$ on $AB$ are parallel to $AC$ , $BC$ and intersect $BC$ , $AC$ at $F$ , $G$, respectively. Prove that the sum of the circumferences of the circles circumscribed around $\vartriangle ADG$ and $\vartriangle BDF$ is equal to the circumference of the circle circumscribed around $\vartriangle ABC$.

2021 Saudi Arabia IMO TST, 5

Tags: geometry , right angle , circumradius , incenter

Let $ABC$ be a non isosceles triangle with incenter $I$ . The circumcircle of the triangle $ABC$ has radius $R$. Let $AL$ be the external angle bisector of $\angle BAC $with $L \in BC$. Let $K$ be the point on perpendicular bisector of $BC$ such that $IL \perp IK$.Prove that $OK=3R$.

2021 Sharygin Geometry Olympiad, 8.5

Tags: geometry , equal circles , circumradius , equal segments

Points $A_1,A_2,A_3,A_4$ are not concyclic, the same for points $B_1,B_2,B_3,B_4$. For all $i, j, k$ the circumradii of triangles $A_iA_jA_k$ and $B_iB_jB_k$ are equal. Can we assert that $A_iA_j=B_iB_j$ for all $i, j$'?

2010 Sharygin Geometry Olympiad, 1

Tags: circumradius , inradius , metric relation , geometry

Let $O, I$ be the circumcenter and the incenter of a right-angled triangle, $R, r$ be the radii of respective circles, $J$ be the reflection of the vertex of the right angle in $I$. Find $OJ$.

II Soros Olympiad 1995 - 96 (Russia), 9.8

Tags: geometry , circumradius

The altitude, angle bisector and median coming from one vertex of the triangle are equal to $\sqrt3$, $2$ and $\sqrt6$, respectively. Find the radius of the circle circumscribed round this triangle.

IV Soros Olympiad 1997 - 98 (Russia), 10.3

Tags: circumradius , geometry

What can angle $B$ of triangle $ABC$ be equal to if it is known that the distance between the feet of the altitudes drawn from vertices $A$ and $C$ is equal to half the radius of the circle circumscribed around this triangle?

2015 Sharygin Geometry Olympiad, P15

Tags: geometry , geometric inequalities , inradius , circumradius

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 $ABC$.

1992 Romania Team Selection Test, 5

Tags: circumradius , sidelenghts , geometry

Let $O$ be the circumcenter of an acute triangle $ABC$. Suppose that the circumradius of the triangle is $R = 2p$, where $p$ is a prime number. The lines $AO,BO,CO$ meet the sides $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. Given that the lengths of $OA_1,OB_1,OC_1$ are positive integers, find the side lengths of the triangle.

2005 Thailand Mathematical Olympiad, 2

Tags: ratio , geometry , circumradius

Let $\vartriangle ABC$ be an acute triangle, and let $A'$ and $B'$ be the feet of altitudes from $A$ to $BC$ and from $B$ to $CA$, respectively; the altitudes intersect at $H$. If $BH$ is equal to the circumradius of $\vartriangle ABC$, find $\frac{A'B}{AB}$ .

2021 Yasinsky Geometry Olympiad, 4

Tags: centroid , circumradius , geometry

$K$ is an arbitrary point inside the acute-angled triangle $ABC$, in which $\angle A = 30^o$. $F$ and $N$ are the points of intersection of the medians in the triangles $AKC$ and $AKB$, respectively . It is known that $FN = q$. Find the radius of the circle circumscribed around the triangle $ABC$. (Grigory Filippovsky)

2010 Estonia Team Selection Test, 3

Tags: inequalities , geometric inequalities , trigonometry , circumradius , geometry

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

2024 Regional Olympiad of Mexico Southeast, 2

Tags: geometry , circumradius , acute triangle , perpendicular

Let $ABC$ be an acute triangle with circumradius $R$. Let $D$ be the midpoint of $BC$ and $F$ the midpoint of $AB$. The perpendicular to $AC$ through $F$ and the perpendicular to $BC$ through $B$ intersect at $N$. Prove that $ND = R$.

Geometry Mathley 2011-12, 6.1

Tags: tangent , geometry , circumradius , excenter , incenter , distance , circumcircle

Show that the circumradius $R$ of a triangle $ABC$ equals the arithmetic mean of the oriented distances from its incenter $I$ and three excenters $I_a,I_b, I_c$ to any tangent $\tau$ to its circumcircle. In other words, if $\delta(P)$ denotes the distance from a point $P$ to $\tau$, then with appropriate choices of signs, we have $$\delta(I) \pm \delta_(I_a) \pm \delta_(I_b) \pm \delta_(I_c) = 4R$$ Luis González