Problem 1847

Let us denote by $A_i = \{a \in A \mid a \equiv i \pmod{2k}\}$ and by $a_i = |A_i|$ for each $i = 0, \dots, 2k-1$. Observe that $A = \bigcup_{i=0}^{2k-1} A_i$, where $A_i \cap A_j = \emptyset$ for $i \neq j$, and therefore $2n = \sum_{i=0}^{2k-1} a_i$.

For any $a \in A_i$, we have
$$P_a = \{b \in A \mid b - a \equiv k \pmod{2k}\} = \{b \in A \mid b \equiv k + i \pmod{2k}\} = A_{k+i},$$
where the index $k + i$ is considered modulo $2k$. Hence, $p_a = a_{k+i}$, for every $a \in A_i$.

Thus,
$$E_A = \sum_{a \in A} p_a^2 = \sum_{i=0}^{2k-1} \sum_{a \in A_i} p_a^2 = \sum_{i=0}^{2k-1} a_i a_{k+i}^2 = \sum_{i=0}^{k-1} a_i a_{k+i}^2 + \sum_{i=k}^{2k-1} a_i a_{k+i}^2.$$
Since the indices are modulo $2k$,
$$\sum_{i=k}^{2k-1} a_i a_{k+i}^2 = \sum_{i=0}^{k-1} a_{k+i} a_i^2,$$
so we get
$$E_A = \sum_{i=0}^{k-1} (a_i a_{k+i}^2 + a_{k+i} a_i^2) = \sum_{i=0}^{k-1} a_i a_{k+i} (a_i + a_{k+i}).$$

We want to maximize $E_A$ under the constraint $\sum_{i=0}^{2k-1} a_i = 2n$.
Note that replacing any two distinct pairs $(a_i, a_{i+k})$ and $(a_j, a_{j+k})$ by $(a_i + a_j, a_{i+k} + a_{j+k})$ and $(0, 0)$ does not decrease the value of $E_A$, because
Therefore, the maximum is achieved when only one pair $(a_i, a_{i+k})$ is nonzero, with $a_i + a_{i+k} = 2n$.

Using the inequality between arithmetic and geometric means, we have $E_A = a_i a_{i+k} (a_i + a_{i+k}) \le \frac{(a_i+a_{i+k})^3}{4} = 2n^3$.
The maximum value $2n^3$ is attained, for example, by any set $A$ having exactly $n$ multiples of $2k$ and $n$ numbers congruent to $k \pmod{2k}$.