MathDB

Initially memory of computer contained a single polynomial

x^2-1

. Every minute computer chooses any polynomial

f(x)

from its memory and writes

f(x^2-1)

and

f(x)^2-1

to it, or chooses any two distinct polynomials

g(x), h(x)

from its memory and writes polynomial

\frac{g(x) + h(x)}{2}

to it (no polynomial is ever erased from its memory). Can it happen that after some time, memory of computer contains

P(x) = \frac{1}{1024}(x^2-1)^{2048} - 1

?
(Proposed by Arsenii Nikolaiev)

Problem Statement