<s>[INST] Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$ 
Prove that $a^7 + b^7 + c^7 \geq a^4b^3 + b^4c^3 + c^4a^3$

Values $a,b,c$ are all positive reals. I tried Muirhead and a few AM $\geq$ GM. 
This problem is equivalent to proving $a^4b^3 + b^4c^3 + c^4a^3 \geq c^2a^5 + b^2c^5+a^2b^5$.
 [/INST] Earth lended me it's energy. Without any assumption on the size of $a,b,c$:
$a^7 + a^7 + a^7 + a^7 + b^7 + b^7 + b^7 \geq 7a^4b^3$
$b^7 + b^7 + b^7 + b^7 + c^7 + c^7 + c^7 \geq 7b^4c^3$
$c^7 + c^7 + c^7 + c^7 + a^7 + a^7 + a^7 \geq 7c^4a^3$
adding up I get what I wanted
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