Question 81
Let $Q_0$ be the set of positive rational numbers. Let $f:Q_0\to\mathbb R$ satisfy:
(i) $f(x)f(y)\ge f(xy)$ for all $x,y\in Q_0$;
(ii) $f(x+y)\ge f(x)+f(y)$ for all $x,y\in Q_0$;
(iii) there exists a rational $a\gt1$ such that $f(a)=a$.
Proof task (stated at the beginning of the source solution): Prove $f(x)=x$ for every positive rational $x$.
(i) $f(x)f(y)\ge f(xy)$ for all $x,y\in Q_0$;
(ii) $f(x+y)\ge f(x)+f(y)$ for all $x,y\in Q_0$;
(iii) there exists a rational $a\gt1$ such that $f(a)=a$.
Proof task (stated at the beginning of the source solution): Prove $f(x)=x$ for every positive rational $x$.