2020/024) Show that if for sides a,b,c of a triangle if for each integer n the sides $a^n,b^n,c^n$ form a triangle then the triangle is isosceles.

Without loss of generality let us assume that a is the longest side. if the triangle is not  isosceles then we have $ a > b$ and $a > c$.

For it to an triangle we need to have $a^n < b^n + c^n$

Or $$(\frac{b}{a})^n +  (\frac{c}{a})^n > 1\cdots(1)$$ for all n

As $b < a$ so $(\frac{b}{a}) < 1$ and  $(\frac{b}{a})^n < 1$ and as n goes to infinity this goes to zero. 

Similarly $(\frac{c}{a})^n < 1$ and as n goes to infinity this goes to zero. 

So the sum goes to zero and hence (1) is not true so $a^n,b^n.c^n$ sides cannot form a triangle

So  either b or c has to be same as a. So the triangle is isosceles.