CMO 1974 Problem 3

Let $$f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}$$ be a polynomial with coefficients satisfying the conditions: $$0\le a_{i}\le a_{0}, i=1,2,\ldots,n.$$ Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial

\begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right) &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*}

Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$ .