> 
> Find all $f : (0, \infty) \rightarrow (0, \infty)$ such that $(f(x))^2
>  \ge f(x + y)(f(x) + y), \forall x, y \gt 0$
> 
> 
> 

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My guess there is no such function but I cannot prove it. 

The most obvious idea is to make $y=x$ but it seem to lead nowhere.

Also, it's easy to show $f$ is strictly decreasing