Olympiad problems

Swiss NMO - geometry, 2005.1

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.

Denmark (Mohr) - geometry, 1992.4

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 Belarus Team Selection Test, 5.2

Tags: Medians , median , inequalities , geometric inequality , geometry

Numbers $a, b, c$ are the length of the medians of some triangle. If $ab + bc + ac = 1$ prove that a) $a^2b + b^2c + c^2a > \frac13$ b) $a^2b + b^2c + c^2a > \frac12$ (I. Bliznets)

Kyiv City MO Juniors 2003+ geometry, 2011.8.41

Tags: geometry , equal angles , Medians

The medians $AL, BM$, and $CN$ are drawn in the triangle $ABC$. Prove that $\angle ANC = \angle ALB$ if and only if $\angle ABM =\angle LAC$. (Veklich Bogdan)

Kyiv City MO Juniors 2003+ geometry, 2019.9.2

Tags: geometry , Medians , perpendicular

In a right triangle $ABC$, the lengths of the legs satisfy the condition: $BC =\sqrt2 AC$. Prove that the medians $AN$ and $CM$ are perpendicular. (Hilko Danilo)

2020 Swedish Mathematical Competition, 2

Tags: geometry , geometric inequality , Medians

The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$.

1994 Swedish Mathematical Competition, 2

Tags: Geometric Inequalities , geometry , Medians

In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge \frac23$.

2023-24 IOQM India, 14

Tags: geometry , coordinate geometry , Medians , Centroid , IOQM

Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroid of the triangle $D E F$. Find the length $G K$.

Durer Math Competition CD Finals - geometry, 2008.C2

Tags: geometry , Medians , geometric inequality , Geometric Inequalities

Given a triangle with sides $a, b, c$ and medians $s_a, s_b, s_c$ respectively. Prove the following inequality: $$a + b + c> s_a + s_b + s_c> \frac34 (a + b + c) $$