<s>[INST] Prove inequality $ (a+b+c)(a^7+b^7+c^7)\ge(a^5+b^5+c^5)(a^3+b^3+c^3)$ for $a,b,c\ge 0$ Prove that:
$$ (a+b+c)(a^7+b^7+c^7)\ge(a^5+b^5+c^5)(a^3+b^3+c^3)$$
I already know that this can be proven using Cauchy Schwarz, but I don't really see how to apply it here. I'm looking for hints. 
 [/INST] Expanding the two sides and cancelling like terms gives
$$a^7b+ab^7+b^7c+bc^7+c^7a+ca^7\geq a^5b^3+a^3b^5+b^5c^3+b^3c^5+c^5a^3+c^3a^5.$$
This follows from Muirhead's inequality, where we set $n=2, a_1=5,a_2=3,b_1=7$, and $b_2=1$.
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