Question 45
Write $\mathbb Q^+$ for the positive rational numbers. Call $f:\mathbb Q^+\to\mathbb Q^+$ elastic if, for every $x,y\in\mathbb Q^+$,
$$f(x)+f(y)\ge4f(x+y).$$
a. Prove that if $f$ is elastic and $x,y,z$ are positive rationals, then $f(x)+f(y)+f(z)\ge8f(x+y+z)$.
b. Does an elastic $f$ exist together with positive rationals $x,y,z$ for which $f(x)+f(y)+f(z)\lt 9f(x+y+z)$?
$$f(x)+f(y)\ge4f(x+y).$$
a. Prove that if $f$ is elastic and $x,y,z$ are positive rationals, then $f(x)+f(y)+f(z)\ge8f(x+y+z)$.
b. Does an elastic $f$ exist together with positive rationals $x,y,z$ for which $f(x)+f(y)+f(z)\lt 9f(x+y+z)$?