<s>[INST] prove this inequality $a^n>b^n+c^n$ We know a,b and c are positive and
 $a^2=b^2+c^2$
How we can conclude this inequality:
$a^n>b^n+c^n$  , $n>2$
I tried Binomial Theorem  but I can't prove this.
Thanks
 [/INST] If $n=2m$ is even, then this is straightforward:
$$ a^n=(a^2)^m=(b^2+c^2)^m> b^{2m}+c^{2m}=b^n+c^n$$
If $n>2$ is odd, then since $n-1$ is even it follows that
$$ a^{n}=a\cdot a^{n-1}\geq a(b^{n-1}+c^{n-1})$$
using either equality if $n-1=2$, or what we have shown above if $n-1>2$.
Finally, $a^2=b^2+c^2$ with $a,b,c>0$ implies that $a>b$ and $a>c$, hence
$$ a(b^{n-1}+c^{n-1})>b\cdot b^{n-1}+c\cdot c^{n-1}=b^n+c^n$$
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