Question 27
Let $k, n \in \mathbb{N}^*$ and let $A$ be an arbitrary set of $2n$ positive integers. For each $a \in A$, denote by $p_a$ the number of elements in the set $P_a = \{b \in A \mid b - a \equiv k \pmod{2k}\}$. Determine the maximum possible value of the sum $E_A = \sum_{a \in A} p_a^2$, when $A$ runs over all subsets of size $2n$ of $\mathbb{N}^*$.
Cristi Săvescu
Cristi Săvescu