<s>[INST] Let $x,y,z>0$ such that $x^2+y^2+z^2=1$. Prove that $x^2yz+y^2zx+z^2xy\le \frac13$ Let $x,y,z>0$ such that $x^2+y^2+z^2=1$. Prove that $x^2yz+y^2zx+z^2xy\le \dfrac13$
My attempt:
I tried using AM-GM and also weighted AM-GM but both seems to be unyielding. So, please help. Thank you.
 [/INST] Apply AM-GM and Cauchy-Schwarz inequalities: 
$LHS =xyz(x+y+z) \leq \dfrac{(x+y+z)^4}{27}\leq \dfrac{(3(x^2+y^2+z^2))^2}{27}=\dfrac{1}{3}$
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