Question 64
Consider the polynomial $P(x) = a_{21}x^{21} + a_{20}x^{20} + \cdots + a_1x + a_0$ with coefficients that are in the interval $[1011, 2021]$. Given that $P(x)$ has an integer root and there exists a positive real number $c$ such that $|a_{k+2} - a_k| \le c$ for all $k \in \{0, 1, \dots, 19\}$.
Part (a). Prove that $P(x)$ has a unique integer root.
Part (b). Prove that $\sum_{k=0}^{10} (a_{2k+1} - a_{2k})^2 \le 440c^2$.
Part (a). Prove that $P(x)$ has a unique integer root.
Part (b). Prove that $\sum_{k=0}^{10} (a_{2k+1} - a_{2k})^2 \le 440c^2$.