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Coefficients of a polynomial

Source: Canada 1974/3

January 13, 2007
algebrapolynomial

Problem Statement

Let f(x)=a0+a1x+a2x2+⋯+anxnf(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 b0,b1,…,b2nb_{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 bn+1≤12(f(1))2b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}.
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