Question 139
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 $x_{0} + x_{1} + \cdots + x_{n} = n$.