<s>[INST] $a,b,c>0$ , $a^a b^b c^c =1$ then $a+b+c \le 3$? Let $a,b,c$ be positive real numbers such that $a^a b^b c^c =1$ , then is it true that $a+b+c \le 3$ ?   
 [/INST] Let $f(x)=x \log x$ , for $x>0$. Clearly $f''(x)=\frac{1}{x}>0~$, so by Jensen's inequality we have
$$\frac{a+b+c}{3}\log\left(\frac{a+b+c}{3}\right)\leq\frac{1}{3}(a\log a+b\log b+c\log c)=
\frac{\log(a^ab^bc^c)}{3}\leq 0$$
it follows that $\log\left(\frac{a+b+c}{3}\right)\leq0$ that is
$a+b+c\leq 3$.
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