4. Given a polynomial of degree 2022 with integer coefficients and leading coefficient 1. What is the maximum number of roots it can have on the interval $(0,1)$?

(A. Kanel-Belov)
Answer: 2021

Solution. If the interval $(0,1)$ contains all 2022 roots of the polynomial, then by Vieta's theorem, the constant term of the polynomial must be equal to their product, and thus will also lie in the interval $(0,1)$ and will not be an integer. We will prove that the polynomial $P(x)=x^{2022}+(1-$ $4042 x)(3-4042 x) \ldots(4041-4042 x)$ can serve as an example. Notice that for all $k=0,1, \ldots, 2021$, the number $P\left(\frac{2 k}{4042}\right)=\left(\frac{2 k}{4042}\right)^{n}+(-1)^{k}(2 k-1)!!(4041-2 k)!!$ is positive for even $k$ and negative for odd $k$. Thus, on the interval $(0,1)$, the polynomial $P(x)$ changes sign at least 2021 times, and therefore has at least 2021 roots.