Problem

Source: 2019 Saint Petersburg

Tags: algebra



A polynomial $f(x)$ of degree $2000$ is given. It's known that $f(x^2-1)$ has exactly $3400$ real roots while $f(1-x^2)$ has exactly $2700$ real roots. Prove that there exist two real roots of $f(x)$ such that the difference between them is less that $0.002$. (А. Солынин)

HIDE: Thanks Thanks to the user Vlados021 for translating the problem.