= Solution
For node $x_i$, let $\ell_i$ be the <Lagrange interpolation polynomial> with $\ell_i(x_j)=\delta_{ij}$. It has degree $n$, so $\ell_i^2$ has degree $2n$. The <Gaussian quadrature> with $n+1$ nodes is exact through degree $2n+1$, hence
$$
a_i=\sum_{j=0}^na_j\ell_i(x_j)^2=\int_{-1}^1\ell_i(x)^2w(x)dx.
$$
The integrand is nonnegative. Since $\ell_i$ is a nonzero <polynomial>, it is nonzero on some open subinterval; the positive weight has positive <integral> there. Consequently $\boxed{a_i>0}$ for every node. This proves the <positive weights of Gaussian quadrature> property without assuming any coefficient sign beforehand.
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