TITLE: prove this inequality $a^n>b^n+c^n$
QUESTION [5 upvotes]: 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

REPLY [2 votes]: $a^2=b^2+c^2\iff 1=(\frac ba)^2+(\frac ca)^2$ and clearly $\frac ba <1$ and $\frac ca <1$. Hence $$1=(\frac ba)^2+(\frac ca)^2>(\frac ba)^3+(\frac ca)^3>(\frac ba)^4+(\frac ca)^4>......>(\frac ba)^n+(\frac ca)^n$$
Thus $$1>(\frac ba)^n+(\frac ca)^n\iff a^n>b^b+c^n$$