NZ-NZMO 2024 Problem 8

Let $a$ , $b$ and $c$ be any positive real numbers. Prove that

$${\frac{a^{2}+b^{2}}{2c}}+{\frac{a^{2}+c^{2}}{2b}}+{\frac{b^{2}+c^{2}}{2a}}\geqslant a+b+c.$$