Question 63
Let $m_1$, $m_2$, $m_3$, $n_1$, $n_2$ and $n_3$ be positive real numbers such that
$$
(m_1 - n_1)(m_2 - n_2)(m_3 - n_3) = m_1 m_2 m_3 - n_1 n_2 n_3.
$$
Prove that
$$
(m_1 + n_1)(m_2 + n_2)(m_3 + n_3) \geq 8m_1 m_2 m_3.
$$
$$
(m_1 - n_1)(m_2 - n_2)(m_3 - n_3) = m_1 m_2 m_3 - n_1 n_2 n_3.
$$
Prove that
$$
(m_1 + n_1)(m_2 + n_2)(m_3 + n_3) \geq 8m_1 m_2 m_3.
$$