\frac{4m^2+25n^2}{2}\geq \sqrt{(2m)^2\cdot (5n)^2}=10mn \Longrightarrow 4m^2+25n^2 \geq 20mn \Longrightarrow \frac{10mn}{4m^2+25n^2+20mn} \leq \frac{10mn}{20mn+20mn}=\frac{10mn}{40mn}= \frac{1}{4}