JBMO 2024 Problem 1

Let $a, b, c$ be positive real numbers such that

$$a^2 + b^2 + c^2 = \frac{1}{4}.$$

Prove that

$$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \leq \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$