XXV OBM

$f(x)$ is a real-valued function defined on the positive reals such that (1) if $x<y$, then $f(x)<f(y)$; (2) $f\left(\frac{2xy}{x+y}\right)=\frac{f(x)+f(y)}{2}$ for all $x$. Show that $f(x)<0$ for some value of $x$.