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2017 OIM Problems/Problem 6

Problem

Let $n > 2$ be an even positive integer and $a_1 < a_2 < \cdots < a_n$ real numbers such that $a_{k+1} -a_k \le 1$ for all $k$ with $1 \le k le n-1$. Let $A$ be the set of pairs $(i, j)$ with $1 \le i < j \le n$ and $j - i$ even, and let $B$ be the set of pairs $(i, j)$ with $1 \le i < j \le n$ and $j - i$ odd. Show that

\[\prod_{(i,j)\in A}^{}(a_j-a_i)>\prod_{(i,j)\in B}^{}(a_j-a_i)\]

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions

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