Question 4
Let $a,b,c$ be the sides of a triangle, with $a+b+c=1$, and let $n\ge2$ be an integer. Show that
$$\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}\lt 1+\frac{\sqrt[n]{2}}2.$$
$$\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}\lt 1+\frac{\sqrt[n]{2}}2.$$