<s>[INST] Maximum value of $\frac{a}{1+bc} + \frac b{1+ac} + \frac{c}{1+ab}$ given $a^2 + b^2 + c^2 = 1$ Given that the constraint of $a, b, c$, for which $a, b, c$ are non-negative real numbers, is $a^2+b^2+c^2=1,$ find the maximum value of $$\frac{a}{(1+bc)}+\frac b{(1+ac)}+\frac{c}{(1+ab)}.$$
For this question I have tried using this geometric method, hopefully it can be logically correct
Do you all have actually better method(s) to solve this problem? 

 [/INST] For non-negative variables we obtain: $$\sum_{cyc}\frac{a}{1+bc}\leq\sum_{cyc}\frac{a\sqrt2}{a+b+c}=\sqrt2$$
because $$2(1+bc)^2\geq(a+b+c)^2$$ it's
$$2+4bc+2b^2c^2\geq1+2(ab+ac+bc)$$ or
$$2b^2c^2+(b+c-a)^2\geq0.$$
The equality occurs fot $c=0$ and $a=b=\frac{1}{\sqrt2},$ which says that we got a maximal value.
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