Problem

Source: 2021 Peru TST D2P3

Tags: functional equation, algebra



Suppose the function $f:[1,+\infty)\to[1,+\infty)$ satisfies the following two conditions: (i) $f(f(x))=x^2$ for any $x\geq 1$; (ii) $f(x)\leq x^2+2021x$ for any $x\geq 1$. 1. Prove that $x<f(x)<x^2$ for any $x\geq 1$. 2. Prove that there exists a function $f$ satisfies the above two conditions and the following one: (iii) There are no real constants $c$ and $A$, such that $0<c<1$, and $\frac{f(x)}{x^2}<c$ for any $x>A$.