Olympiad problems

2015 Belarus Team Selection Test, 2

The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$. I. Gorodnin

2000 Belarus Team Selection Test, 5.1

Tags: Geometric Inequalities , inequalities , Medians , angle bisector

Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$). If $BC = a, AM = m_a, AL = l_a$, prove the inequalities: (a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$ (b) $2l_a \le a\cot \frac{a}{2} $.

1989 Mexico National Olympiad, 1

Tags: geometry , Medians , area of a triangle , perpendicular

In a triangle $ABC$ the area is $18$, the length $AB$ is $5$, and the medians from $A$ and $B$ are orthogonal. Find the lengths of the sides $BC,AC$.

2009 Oral Moscow Geometry Olympiad, 4

Tags: geometry , concurrency , concurrent , Medians , diameter

Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let $C_1$ be the intersection point of the circles built on the medians $AM_1$ and $BM_2$, which is more distant from the vertex $C$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ intersect at one point. (D. Tereshin)

1959 Czech and Slovak Olympiad III A, 1

Tags: construction , Triangle , Medians

Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.

2006 Thailand Mathematical Olympiad, 3

Tags: geometry , Medians

The three medians of a triangle has lengths $3, 4, 5$. What is the length of the shortest side of this triangle?

2005 Switzerland - Final Round, 1

Tags: geometry , Equilateral , Medians , Centroid , Concyclic , cyclic quadrilateral

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

2014 Saudi Arabia GMO TST, 1

Tags: Concyclic , geometry , Cyclic , Medians

Let $ABC$ be a triangle with $\angle A < \angle B \le \angle C$, $M$ and $N$ the midpoints of sides $CA$ and $AB$, respectively, and $P$ and $Q$ the projections of $B$ and $C$ on the medians $CN$ and $BM$, respectively. Prove that the quadrilateral $MNPQ$ is cyclic.

2012 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , median , Medians , area of a triangle , maximum

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

Kyiv City MO Juniors 2003+ geometry, 2017.9.51

Tags: geometry , tangential quadrilateral , Medians , isosceles

In the triangle $ABC$, the medians $BB_1$ and $CC_1$, which intersect at the point $M$, are drawn. Prove that a circle can be inscribed in the quadrilateral $AC_1MB_1$ if and only if $AB = AC$.

2019 Ramnicean Hope, 2

Tags: geometry , circumcircle , Medians , vectorial geometry

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

1940 Eotvos Mathematical Competition, 3

Tags: geometry , similar , Medians , median , triangle inequality

(a) Prove that for any triangle $H_1$, there exists a triangle $H_2$ whose side lengths are equal to the lengths of the medians of $H_1$. (b) If $H_3$ is the triangle whose side lengths are equal to the lengths of the medians of $H_2$, prove that $H_1$ and $H_3$ are similar.

2007 Sharygin Geometry Olympiad, 5

Tags: geometry , Centroid , Medians , acute

Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$. Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.

1959 Poland - Second Round, 2

Tags: geometry , similar , Medians

What relationship between the sides of a triangle makes it similar to the triangle formed by its medians?

2017 QEDMO 15th, 8

Tags: geometry , Medians , triangle inequality

Let $ABC$ be a triangle of area $1$ with medians $s_a, s_b,s_c$. Show that there is a triangle whose sides are the same length as $s_a, s_b$, and $s_c$, and determine its area.

1992 Denmark MO - Mohr Contest, 4

Tags: geometry , Medians , geometric inequality

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

2010 Gheorghe Vranceanu, 1

Tags: geometry , Medians , concurrence

Let $ A_1,B_1,C_1 $ be the middlepoints of the sides of a triangle $ ABC $ and let $ A_2,B_2,C_2 $ be on the middle of the paths $ CAB,ABC,BCA, $ respectively. Prove that $ A_1A_2,B_1B_2,C_1C_2 $ are concurrent.

2004 All-Russian Olympiad Regional Round, 9.2

Tags: geometry , isosceles , Medians

In triangle $ABC$, medians $AA'$, $BB'$, $CC'$ are extended until they intersect with the circumcircle at points $A_0$, $B_0$, $C_0$, respectively. It is known that the intersection point M of the medians of triangle $ABC$ divides the segment $AA_0$ in half. Prove that the triangle $A_0B_0C_0$ is isosceles.

2016 Sharygin Geometry Olympiad, 1

Tags: geometry , Medians , right triangle , altitude

An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle. by Yu.Blinkov

1998 Italy TST, 2

Tags: geometry , Equilateral , Medians , angle bisector , altitude

In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.

2011 Saudi Arabia Pre-TST, 2.4

Tags: similar triangles , geometry , Medians , median

Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that: (a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$. (b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.

2000 Belarus Team Selection Test, 3.1

Tags: Medians , geometry

In a triangle $ABC$, let $a = BC, b = AC$ and let $m_a,m_b$ be the corresponding medians. Find all real numbers $k$ for which the equality $m_a+ka = m_b +kb$ implies that $a = b$.

1999 All-Russian Olympiad Regional Round, 8.3

Tags: geometry , Medians , inscribed , ratio

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

2014 Austria Beginners' Competition, 4

Tags: geometry , Medians , perpendicular

Consider a triangle $ABC$. The midpoints of the sides $BC, CA$, and $AB$ are denoted by $D, E$, and $F$, respectively. Assume that the median $AD$ is perpendicular to the median $BE$ and that their lengths are given by $AD = 18$ and $BE = 13.5$. Compute the length of the third median $CF$. (K. Czakler, Vienna)

1998 Italy TST, 2

Tags: geometry , Equilateral , Medians , angle bisector , altitude

In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.