Find max: $M=\frac{a}{b^2+c^2+a}+\frac{b}{c^2+a^2+b}+\frac{c}{a^2+b^2+c}$ For $a,b,c>0$ and $abc=1$, find the maximum of
$$M=\frac{a}{b^2+c^2+a}+\frac{b}{c^2+a^2+b}+\frac{c}{a^2+b^2+c}.$$

Take the first term:$$\frac{a}{b^2+c^2+a}=\frac{1}{\frac{1}{a^2}(\frac{b}{c}+\frac{c}{b})+1}$$
Minimum value of $\frac{1}{a^2}(\frac{b}{c}+\frac{c}{b})$ is $\frac{2}{a^2}$ which it takes when $b=c$. By the symmetry of the expression it is maximum when $a=b=c$ and the maximum value is $1$.
