Question 5
Let $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots+a_1x+a_0$, where all coefficients belong to $[1011,2021]$. Assume that $P$ has an integer root and that $c\gt 0$ satisfies $|a_{k+2}-a_k|\le c$ for every $k\in\{0,1,\ldots,19\}$.
a) Prove that $P$ has exactly one integer root.
b) Prove that $\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\le440c^2$.
a) Prove that $P$ has exactly one integer root.
b) Prove that $\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\le440c^2$.