Let $d \in \mathbb{N}$, and $x_1, \ldots, x_d$, non-negative real numbers, where $x_1 + \ldots + x_d = 1$.
For each $ i \in \{1, \ldots, d\} $, let
$$f(i) = \displaystyle \sum_{(l_1,\ldots, l_d) \in \mathbb{N}^d} \displaystyle \binom{l_1 + \ldots + l_d}{l_1, \ldots, l_d}^2 x_1^{2l_1} \ldots x_d^{2l_d} \frac{2l_i}{x_i} .$$
Prove that $f(i) = f(j),~ \forall i, j \in \{1, \ldots, d\}$ if, and only if, $x_i = x_j, ~ \forall i, j \in \{1, \ldots, d\}.$