INMO 1988 Problem 1

Let $m_1,m_2,m_3,\dots,m_n$ be a rearrangement of the numbers $1,2,\dots,n$ . Suppose that $n$ is odd. Prove that the product $$\left(m_1-1\right)\left(m_2-2\right)\cdots \left(m_n-n\right)$$ is an even integer.