DE-MO 2022 Problem 6

Consider functions $f$ satisfying the following four conditions: (1) $f$ is real-valued and defined for all real numbers. (2) For any two real numbers $x$ and $y$ we have $f(xy)=f(x)f(y)$ . (3) For any two real numbers $x$ and $y$ we have $f(x+y) \le 2(f(x)+f(y))$ . (4) We have $f(2)=4$ .

Prove that: a) There is a function $f$ with $f(3)=9$ satisfying the four conditions. b) For any function $f$ satisfying the four conditions, we have $f(3) \le 9$ .