Question 2
Let n be a positive integer. Find the minimum possible value of
$$ S=2^{0}x_{0}^{2}+2^{1}x_{1}^{2}+\cdots+2^{n}x_{n}^{2}, $$
where $x_{0}, x_{1}, \ldots, x_{n}$ are nonnegative integers such that $x_{0} + x_{1} + \cdots + x_{n} = n$.
$$ S=2^{0}x_{0}^{2}+2^{1}x_{1}^{2}+\cdots+2^{n}x_{n}^{2}, $$
where $x_{0}, x_{1}, \ldots, x_{n}$ are nonnegative integers such that $x_{0} + x_{1} + \cdots + x_{n} = n$.