109. Find all real numbers $x, y, z, w$ such that
$$x+y+z=\frac{3}{2}, \sqrt{4 x-1}+\sqrt{4 y-1}+\sqrt{4 z-1} \geqslant 2+3 \sqrt{w-2}$$
(2007-2008 Hungarian Mathematical Olympiad)
109. By the mean inequality,
$$\frac{\sqrt{4 x-1}+\sqrt{4 y-1}+\sqrt{4 z-1}}{3} \leqslant \sqrt{\frac{4 x-1+4 y-1+4 z-1}{3}}=1$$

Equality holds if and only if $x=y=z=\frac{1}{2}$. Therefore, $2+3 \sqrt{w-2} \leqslant \sqrt{4 x-1}+$ $\sqrt{4 y-1}+\sqrt{4 z-1} \leqslant 3$, hence, $\sqrt{w-2} \leqslant 0$. Thus, $w=2$, and all the equalities hold. At this point, $x=y=z=\frac{1}{2}, w=2$.