Question 96
Let $k$ be a positive integer, and let $(a_{1}, a_{2}, \ldots, a_{2k})$ and $(b_{1}, b_{2}, \ldots, b_{2k})$ be two sequences of real numbers such that $\frac{1}{2} \leq a_{1} \leq \cdots \leq a_{2k}$ and $\frac{1}{2} \leq b_{1} \leq \cdots \leq b_{2k}$. Let $M$ and $m$ be the maximum and minimum respectively of
$$
(a_{1}+c_{1})(a_{2}+c_{2}) \cdots (a_{2k}+c_{2k})
$$
as $(c_{1}, \ldots, c_{2k})$ ranges through all possible permutations of $(b_{1}, \ldots, b_{2k})$. Prove that
$$
M-m \geq k(a_{k}-a_{k+1})(b_{k}-b_{k+1}) .
$$
$$
(a_{1}+c_{1})(a_{2}+c_{2}) \cdots (a_{2k}+c_{2k})
$$
as $(c_{1}, \ldots, c_{2k})$ ranges through all possible permutations of $(b_{1}, \ldots, b_{2k})$. Prove that
$$
M-m \geq k(a_{k}-a_{k+1})(b_{k}-b_{k+1}) .
$$