1. Given complex numbers $z$ and $\omega$ satisfy the following two conditions:
(1) $z+\omega+3=0$;
(2) $|z|, 2, |\omega|$ form an arithmetic sequence.
Is there a maximum value for $\cos (\arg z - \arg \omega)$? If so, find it.
1. From the given, the vectors corresponding to $z, \omega, 3$ can form a triangle with vertices connected head to tail, or three collinear segments, and $|z|+|\omega|=4$. Thus, the problem is equivalent to finding $\cos \left(180^{\circ}-\alpha\right)$ (i.e., $\left.\cos (\arg z-\arg \omega)\right)$ in $\triangle A O B$, given the side opposite to $\alpha$ and the sum of the other two sides. It is easy to see that $\cos (\arg z-\arg \omega)=1-\frac{7}{2|z| \cdot|\omega|} \leqslant 1-\frac{7}{2[(|z|+|\omega|) / 2]^{2}}=\frac{1}{8}$, hence when and only when $|z|=$ $|\omega|=2$, $[\cos (\arg z-\arg \omega)]_{\text {max }}=\frac{1}{8}$.