Question 46
Find all integers $n \ge 2$ for which there exist the real numbers $a_k$, $1 \le k \le n$, which are satisfying the following conditions:
$$
\sum_{k=1}^{n} a_k = 0, \quad \sum_{k=1}^{n} a_k^2 = 1 \text{ and } \sqrt{n} \cdot \left( \sum_{k=1}^{n} a_k^3 \right) = 2(b\sqrt{n} - 1), \text{ where } b = \max_{1 \le k \le n} \{a_k\}.
$$
$$
\sum_{k=1}^{n} a_k = 0, \quad \sum_{k=1}^{n} a_k^2 = 1 \text{ and } \sqrt{n} \cdot \left( \sum_{k=1}^{n} a_k^3 \right) = 2(b\sqrt{n} - 1), \text{ where } b = \max_{1 \le k \le n} \{a_k\}.
$$