INMO 2006 Problem 5

In a cyclic quadrilateral $A B C D, A B=a, B C=b, C D=c, \angle A B C=120^{\circ}$, and $\angle A B D=$ $30^{\circ}$. Prove that

(i) $c \geq a+b$;

(ii) $|\sqrt{c+a}-\sqrt{c+b}|=\sqrt{c-a-b}$.