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a bound for f(i)

Source: Iran 3rd round 2011-algebra exam-p3

September 7, 2011
algebrapolynomialsearchalgebra proposed

Problem Statement

We define the polynomial f(x)f(x) in R[x]\mathbb R[x] as follows: f(x)=xn+an−2xn−2+an−3xn−3+.....+a1x+a0f(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+.....+a_1x+a_0 Prove that there exists an ii in the set {1,....,n}\{1,....,n\} such that we have ∣f(i)∣≥n!(ni)|f(i)|\ge \frac{n!}{\dbinom{n}{i}}.
proposed by Mohammadmahdi Yazdi
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