MS_Kekas wrote: 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) No. Easy to prove that for $u=\frac{1+\sqrt 5}2$, root of $u^2-1=u$, any function in memory is such that $f(u)=u$ And easy to prove that $P(u)=u$ is wrong : it would be $u^{2048}=1024(u+1)$ which is wrong since $u^{2048}>2^{1024}>2^{12}=1024(3+1)>1024(u+1)$