Question 162
Let $n \in \mathbb{N}^*, n \geq 2$. For real numbers $a_{1}, a_{2}, \ldots, a_{n}$, let $S_{0}=1$ and
$$
S_{k}=\sum_{1 \leq i_{1}\lt i_{2}\lt \ldots\lt i_{k} \leq n} a_{i_{1}} a_{i_{2}} \ldots a_{i_{k}}
$$
be the sum of all products of $k$ numbers chosen among $a_{1}, a_{2}, \ldots, a_{n}$, $k \in \{1,2, \ldots, n\}$.
Determine the number of $n$-tuples $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ for which the following relation holds:
$$
\left(S_{n}-S_{n-2}+S_{n-4}-\ldots\right)^{2}+\left(S_{n-1}-S_{n-3}+S_{n-5}-\ldots\right)^{2}=2^{n} S_{n}
$$
$$
S_{k}=\sum_{1 \leq i_{1}\lt i_{2}\lt \ldots\lt i_{k} \leq n} a_{i_{1}} a_{i_{2}} \ldots a_{i_{k}}
$$
be the sum of all products of $k$ numbers chosen among $a_{1}, a_{2}, \ldots, a_{n}$, $k \in \{1,2, \ldots, n\}$.
Determine the number of $n$-tuples $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ for which the following relation holds:
$$
\left(S_{n}-S_{n-2}+S_{n-4}-\ldots\right)^{2}+\left(S_{n-1}-S_{n-3}+S_{n-5}-\ldots\right)^{2}=2^{n} S_{n}
$$