 Solution

For each node define its Lagrange interpolation cardinal polynomial ${\ell }_i(t)={\prod }_{j\ne i}(t-x_j)/(x_i-x_j)$. Its square has degree $2n-2<2n$ and nodal values ${\ell }_i(x_j{)}^2={\delta }_{ij}$. Exactness therefore gives $a_i={\int }_{-1}^1{\ell }_i(t{)}^{2}dt$. The polynomial is nonzero, so continuity makes this integral strictly positive. Thus $\boxed{a_i>0}$ for every node.