Question 70
Let $a$, $b$ and $c$ be the side-lengths of a triangle with perimeter $1$. Prove that
$$
\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{c^2 + a^2} \lt 1 + \frac{\sqrt{2}}{2}.
$$
(APMO 2003)
Supplemental illustration (AI-generated):
$$
\sqrt{a^2 + b^2} + \sqrt{b^2 + c^2} + \sqrt{c^2 + a^2} \lt 1 + \frac{\sqrt{2}}{2}.
$$
(APMO 2003)
Supplemental illustration (AI-generated):