Question 2
Let $a$, $b$, $c$ be the sides of a triangle, with $a + b + c = 1$, and let $n \geq 2$ be an integer. Show that
$$
\sqrt[n]{a^{n} + b^{n}} + \sqrt[n]{b^{n} + c^{n}} + \sqrt[n]{c^{n} + a^{n}} \lt 1 + \frac{\sqrt[n]{2}}{2}
$$
$$
\sqrt[n]{a^{n} + b^{n}} + \sqrt[n]{b^{n} + c^{n}} + \sqrt[n]{c^{n} + a^{n}} \lt 1 + \frac{\sqrt[n]{2}}{2}
$$