2019/012) if a,b,c are sides of a triangle with the the property that $a^n,b^n,c^n$ are sides of a triangle for each positive integer n then prove that the triangle is isosceles

Without loss of generality let us assume that $a > b>c$

So $a^n < b^n + c^n$
Or $1 < (\frac{b}{a})^n + (\frac{c}{a})^n$
If $a>b > c$ then $(\frac{b}{a})^n = 0$ and $(\frac{c}{a})^n =0$ when n goes to infinite
So $1 < 0$ which is contradiction
So b has to be same as a for the sum to be 1 in the limit
Or the triangle is isosceles

Proved