Question 20
(a) For which triangles with sides of length $a$, $b$ and $c$ do the inequalities $a^2 + b^2 \gt c^2$, $b^2 + c^2 \gt a^2$ and $a^2 + c^2 \gt b^2$ hold (along with the usual triangle inequalities $a+b\gt c$, $b+c\gt a$ and $c+a\gt b$)?
(b) For which triangles with sides of length $a$, $b$ and $c$ do the inequalities $a^n + b^n \gt c^n$, $b^n + c^n \gt a^n$ and $a^n + c^n \gt b^n$ hold for all positive integers $n$ (along with the usual triangle inequalities $a+b\gt c$, $b+c\gt a$ and $c+a\gt b$)?
(b) For which triangles with sides of length $a$, $b$ and $c$ do the inequalities $a^n + b^n \gt c^n$, $b^n + c^n \gt a^n$ and $a^n + c^n \gt b^n$ hold for all positive integers $n$ (along with the usual triangle inequalities $a+b\gt c$, $b+c\gt a$ and $c+a\gt b$)?