<s>[INST] Proving an inequality using AM-GM 
Let $ a,b,c > 0$ such that $ abc = 1$. Prove that 
  $$\frac {ab}{a^2 + b^2 + \sqrt {c}} + \frac {bc}{b^2 + c^2 + \sqrt {a}} + \frac {ca}{c^2 + a^2 + \sqrt {b}}\le 1.$$

$$\frac{ab}{a^2+b^2+\sqrt{c}}=\frac{1}{\frac{a}{b}+\frac{b}{c}+\sqrt{c^3}}$$
(AM-GM): $$\frac{a}{b}+\frac{b}{a}+\sqrt{c^3}\geq 3\sqrt{c} $$ 
$$\frac{1}{\frac{a}{b}+\frac{b}{a}+\sqrt{c^3}}\leq \frac{1}{3\sqrt{c}}$$
Analogous: $$\frac{1}{\frac{b}{c}+\frac{c}{b}+\sqrt{a^3}} \leq \frac{1}{3\sqrt a}$$
$$\frac{1}{\frac{a}{c}+\frac{c}{a}+\sqrt{b^3}}\leq \frac{1}{3\sqrt b}$$
$$LHS \leq \frac{1}{3}+\frac{1}{3}+\frac{1}{3} =1$$
Am I right?
 [/INST] Here is an easier way to prove the inequality. Just prove:
$$\frac {ab}{a^2 + b^2 + \sqrt {c}} \leq \frac {a+b}{2a + 2b + 2{c}} $$ and you are done! It is equivalent to $$a^2b+ab^2+2\leq a^3+b^3+ a\sqrt{c}+b\sqrt{c}$$ which is easy to see since $2\leq a\sqrt{c}+b\sqrt{c}$ and $ a^2b+ab^2\leq a^3+b^3$ are true by  Am-Gm inequality

Edit to a comment. I tried to find such a number $k$ that a following is true:
$$\frac {ab}{a^2 + b^2 + \sqrt {c}} \leq \frac {a^k+b^k}{2a^k + 2b^k + 2c^k} $$
Playing with $a,b,c$ (I put $a=b=x$ and $c=1/x^2$) a find that this is possible only at $k=1$. This is a method I saw when trying to solve: https://artofproblemsolving.com/community/c6h17451p119168
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