Question 86
Let $\mathbb Q_{\gt0}$ be the positive rational numbers. Suppose $f:\mathbb Q_{\gt0}\to\mathbb R$ satisfies:
(i) $f(x)f(y)\geq f(xy)$ for all positive rationals $x,y$;
(ii) $f(x+y)\geq f(x)+f(y)$ for all positive rationals $x,y$;
(iii) there is a rational $a\gt1$ such that $f(a)=a$.
Prove $f(x)=x$ for every positive rational $x$.
(i) $f(x)f(y)\geq f(xy)$ for all positive rationals $x,y$;
(ii) $f(x+y)\geq f(x)+f(y)$ for all positive rationals $x,y$;
(iii) there is a rational $a\gt1$ such that $f(a)=a$.
Prove $f(x)=x$ for every positive rational $x$.