<s>[INST] inequalites of an acute triangle angles $ 180^{180}*a^b*b^c*c^a \le (a^2+b^2+c^2)^{180} $ If $a,b,c$ are an acute angle of triangle the prove that
$ 180^{180}*a^b*b^c*c^a \le (a^2+b^2+c^2)^{180} $
No idea
 [/INST] To solve this you should know (this fact)[https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means]
Hint:
$180 = a + b + c$
Then we can use the fact that:
$\frac{a\cdot b + b\cdot c + c\cdot a}{a+b+c} \ge (a+b+c)(a^b\cdot b^c \cdot c^a)^{\frac{1}{a+b+c}} $
After you understood how to use all this facts you will have the inequality:
$\frac{a\cdot b + b\cdot c + a\cdot c}{a+b+c} \le a^2 + b^2 + c^2$
That much more easy to prove.
One more fact that might be helpful:
$\frac{a+b}{2} \le (\frac{a^2 + b^2}{2})^{\frac{1}{2}}$ 
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