VMO 2021 Problem 5

Let the polynomial $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+a_0$ where $1011\leq a_i\leq 2021$ for all $i=0,1,2,...,21.$ Given that $P(x)$ has an integer root and there exists an positive real number $c$ such that $|a_{k+2}-a_k|\leq c$ for all $k=0,1,...,19.$ a) Prove that $P(x)$ has an only integer root.

b) Prove that $$\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\leq 440c^2.$$