The solution is something like 22 or 23 people, I can't remember right now.
You must calculate for what $\,n\,$ you get
$$\prod_{k=1}^\infty\left(1-\frac{n}{365}\right)\geq\frac{1}{2}$$$$\prod_{k=1}^n\left(1-\frac{k}{365}\right)\geq\frac{1}{2}$$
Well, now just solve the above...(there's a way to simplify it using the binomial coefficient)