Baltic Way 1995 · Problem 8

Solution:

Squaring both sides of the given inequalities we get

$$\left\{ \begin{array}{l} a^{2} \geq (b+c)^{2} \\ b^{2} \geq (c+a)^{2} \\ c^{2} \geq (a+b)^{2} \end{array} \right.$$

Adding these three inequalities and rearranging, we get $(a+b+c)^{2} \leq 0$. Clearly equality must hold, and we have $a+b+c=0$.