Let $-1 \leqslant x_{1}<x_{2}<\ldots<x_{n} \leqslant 1$ and let $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers such that

$$\int_{-1}^{1} p(t) d t=\sum_{i=1}^{n} a_{i} p\left(x_{i}\right)$$

for every polynomial $p$ of degree less than $2 n$. Prove the following three facts.

(i) $a_{i}>0$ for every $i$.

(ii) $\sum_{i=1}^{n} a_{i}=2$.

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

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