Problem

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

Tags: algebra, polynomial, search, algebra proposed



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