Prove that all for all positive integers $m,n $: $\frac{1}{\sqrt[n]{m}}+\frac{1}{\sqrt[m]{n}}>1$
Solution 1:
Turning my earlier comment into an answer: The result follows by setting $x = \frac1{m}$ and $y = \frac1{n}$ in the inequality $x^y + y^x > 1$.