Question 40
Let $a$, $b$, $c$ be the lengths of the sides of a given triangle and $m_a$, $m_b$, $m_c$ be the lengths of the corresponding medians. Prove that:
$$
m_a \left(\frac{b}{a} - 1\right) \left(\frac{c}{a} - 1\right) + m_b \left(\frac{a}{b} - 1\right) \left(\frac{c}{b} - 1\right) + m_c \left(\frac{a}{c} - 1\right) \left(\frac{b}{c} - 1\right) \geq 0.
$$
$$
m_a \left(\frac{b}{a} - 1\right) \left(\frac{c}{a} - 1\right) + m_b \left(\frac{a}{b} - 1\right) \left(\frac{c}{b} - 1\right) + m_c \left(\frac{a}{c} - 1\right) \left(\frac{b}{c} - 1\right) \geq 0.
$$