Question 5
A5 (UKR). Let $\mathbb R$ be the set of real numbers. Does there exist a function $f:\mathbb R\to\mathbb R$ that simultaneously satisfies the following three conditions?
(a) There is a positive number $M$ such that $-M\le f(x)\le M$ for all $x$.
(b) $f(1)=1$.
(c) If $x\ne0$, then
$$f\left(x+\frac1{x^2}\right)=f(x)+\left[f\left(\frac1x\right)\right]^2.$$
(a) There is a positive number $M$ such that $-M\le f(x)\le M$ for all $x$.
(b) $f(1)=1$.
(c) If $x\ne0$, then
$$f\left(x+\frac1{x^2}\right)=f(x)+\left[f\left(\frac1x\right)\right]^2.$$