Question 13
Let $n$ be a positive integer, and let $(a_1, a_2, \ldots, a_n)$ be any $n$-tuple of distinct integers. An inversion is a pair $(i, j)$ of integers satisfying $1 \leq i \lt j \leq n$ and $a_i \gt a_j$. An inversion $(i, j)$ is called $\text{odd}$ if $j - i$ is odd. Let $A$ be the number of all inversions, and let $B$ be the number of odd inversions.
(a) Prove that $A \leq 5B$ for all $n$ and all $n$-tuples $(a_1, a_2, \ldots, a_n)$.
(b) Prove that there exist an n and an n-tuple $ (a_1, a_2, \ldots, a_n) $ for which A > 4.99B.
(a) Prove that $A \leq 5B$ for all $n$ and all $n$-tuples $(a_1, a_2, \ldots, a_n)$.
(b) Prove that there exist an n and an n-tuple $ (a_1, a_2, \ldots, a_n) $ for which A > 4.99B.