Let $A_{1}, A_{2}, \ldots, A_{n}$ be real numbers and suppose that $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ are distinct. Suppose that the formula

$$\int_{-1}^{1} p(x) d x=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

is valid for every polynomial $p$ of degree $\leqslant 2 n-1$. Prove the following:

(i) $A_{j}>0$ for each $j=1,2, \ldots, n$.

(ii) $\sum_{j=1}^{n} A_{j}=2$.

(iii) $x_{1}, x_{2}, \ldots, x_{n}$ are the roots of the Legendre polynomial of degree $n$.

[You may assume standard orthogonality properties of the Legendre polynomials.]