Happy new year to everybody. Ok, see if you like this problem as much as I do:

Let $\displaystyle p(x)=x^n + a_1x^{n-1} + \cdots +a_{n-1}x + a_n \in \mathbb{C}[x],$ and suppose that $\displaystyle \forall k \leq n: \ \ |a_k| \leq k^2 - \frac{4}{3}k + 1.$ Prove that if $\displaystyle p(\alpha)=0,$ then $\displaystyle |\alpha| < 3.$

Source: Alex Lupas

Note: The problem has a simple generalization.