For classical inequalities lovers

Algebra Level 4

Let \(a, b\) and \(c\) positive real numbers such that \(\dfrac{1}{a^2} +\dfrac{1}{b^2} +\dfrac{1}{c^2}=9\). If \(M\) is the maximum value of the following expression \[ \frac{1}{2a+b} +\frac{1}{2b+c} +\frac{1}{2c+a}, \] enter \(M^2\) as your answer.