Question 51
Let $a$, $b$, $c$ and $m_a$, $m_b$, $m_c$ be respectively the lengths of the sides and the medians of an acute-angled triangle $ABC$. Prove that
$$
\frac{m_a^2}{b^2 + c^2 - a^2} + \frac{m_b^2}{c^2 + a^2 - b^2} + \frac{m_c^2}{a^2 + b^2 - c^2} \geq \frac{9}{4}
$$
$$
\frac{m_a^2}{b^2 + c^2 - a^2} + \frac{m_b^2}{c^2 + a^2 - b^2} + \frac{m_c^2}{a^2 + b^2 - c^2} \geq \frac{9}{4}
$$