Interesting 3 Variable Inequality for Real Numbers I recently saw an olympiad style inequality that seemed very difficult. I tried to use elementary inequalities such as AM-GM or Cauchy-Schwarz, but neither have helped in making significant progress. Could anyone provide a rigorous proof, preferably using more elementary inequalities?
Problem: "Prove the inequality $\frac{1}{y(x+y)}+\frac{1}{z(y+z)}+\frac{1}{x(z+x)}\geq\frac{27}{2(x+y+z)^2}$ if $x,y,z$ are positive reals."

This follows almost immediately by a direct application of Hölder's inequality (generalised Cauchy-Schwarz):
$$\begin{aligned} \left ( \sum \limits_{\text{cyc}} \frac{1}{y(x+y)} \right ) (y + z + x) ((x+y) + (y+z) + (z+x)) &\geq (1^{\frac{1}{3}} + 1^{\frac{1}{3}} + 1^{\frac{1}{3}})^{3}\\
\iff \sum \limits_{\text{cyc}} \frac{1}{y(x+y)} &\geq \frac{27}{(x+y+z)(2x+2y+2z)}\\
&= \frac{27}{2(x+y+z)^2}\\
\end{aligned}$$
