Olympiad problems

2014 Dutch Mathematical Olympiad, 5

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 $2$?

IV Soros Olympiad 1997 - 98 (Russia), 10.3

Tags: geometry , circumradius

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?

2005 Spain Mathematical Olympiad, 3

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

1992 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: geometry , circumradius , circumcircle

Let $D$ be the center of the circumcircle of the acute triangle $ABC$. If the circumcircle of triangle $ADB$ intersects $AC$ (or its extension) at $M$ and also $BC$ (or its extension) at $N$, show that the radii of the circumcircles of $\triangle ADB$ and $\triangle MNC$ are equal.

1993 Spain Mathematical Olympiad, 3

Tags: circumradius , geometric inequality , inradius , geometry

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

2024 Regional Olympiad of Mexico Southeast, 2

Tags: acute triangle , circumradius , perpendicular , geometry

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 $ND = R$.

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 $ABC$.

2015 Balkan MO Shortlist, A2

Tags: circumradius , semiperimeter , geometric inequality , inequalities , inradius , geometry

Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that: $$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} \leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$ (Albania)

1992 Swedish Mathematical Competition, 5

Tags: right triangle , geometry , right angle , circumradius

A triangle has sides $a, b, c$ with longest side $c$, and circumradius $R$. Show that if $a^2 + b^2 = 2cR$, then the triangle is right-angled.

2016 Oral Moscow Geometry Olympiad, 6

Tags: circumcircle , circumradius , ratio , symmetry , geometry

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 $O_AO_BO_C$.

1988 Tournament Of Towns, (195) 2

Tags: orthocenter , circumradius , geometry , circumcircle

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.

1930 Eotvos Mathematical Competition, 3

Tags: circumradius , geometric inequalities , geometry

Inside an acute triangle $ABC$ is a point $P$ that is not the circumcenter. Prove that among the segments $AP$, $BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.

2007 Estonia Team Selection Test, 2

Tags: altitude , geometric inequalities , circumradius , incircle , circumcircle , inradius , geometry

Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$

2007 Estonia Team Selection Test, 2

Tags: altitude , circumradius , geometric inequalities , circumcircle , incircle , inradius , geometry

Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$

2009 Sharygin Geometry Olympiad, 2

Tags: circumradius , geometry , cyclic quadrilateral

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)

1985 All Soviet Union Mathematical Olympiad, 412

Tags: circumradius , circumcircle , equal circles , 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 $R$.

1991 All Soviet Union Mathematical Olympiad, 547

Tags: circumradius , equal circles , circumcircle , geometry

$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.

2017 Argentina National Math Olympiad Level 2, 5

Tags: circumradius , angle bisector , equal segments , 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.

2018 JBMO Shortlist, G4

Tags: geometry , incenter , circumradius , geometric inequality , inequalities

Let $ABC$ be a triangle with side-lengths $a, b, c$, inscribed in a circle with radius $R$ and let $I$ be ir's incenter. Let $P_1, P_2$ and $P_3$ be the areas of the triangles $ABI, BCI$ and $CAI$, respectively. Prove that $$\frac{R^4}{P_1^2}+\frac{R^4}{P_2^2}+\frac{R^4}{P_3^2}\ge 16$$

2007 Swedish Mathematical Competition, 3

Tags: circumradius , trigonometry , geometry

Let $\alpha$, $\beta$, $\gamma$ be the angles of a triangle. If $a$, $b$, $c$ are the side length of the triangle and $R$ is the circumradius, show that \[ \cot \alpha + \cot \beta +\cot \gamma =\frac{R\left(a^2+b^2+c^2\right)}{abc} \]

2007 Sharygin Geometry Olympiad, 17

Tags: circumradius , geometry , circumcircle

What triangles can be cut into three triangles having equal radii of circumcircles?

2005 Mexico National Olympiad, 1

Tags: circumradius , geometry , circumcircle

Let $O$ be the center of the circumcircle of an acute triangle $ABC$, let $P$ be any point inside the segment $BC$. Suppose the circumcircle of triangle $BPO$ intersects the segment $AB$ at point $R$ and the circumcircle of triangle $COP$ intersects $CA$ at point $Q$. (i) Consider the triangle $PQR$, show that it is similar to triangle $ABC$ and that $O$ is its orthocenter. (ii) Show that the circumcircles of triangles $BPO$, $COP$, $PQR$ have the same radius.

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 $OK=3R$.

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.

2010 Estonia Team Selection Test, 3

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

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?