JBMO 2024 | Problem 1 - 27 June 2024

Problem 1.

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

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

Prove that

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