Smallest class where a shared birthday is likely

Take a year of $365$ equally likely birthdays. What is the smallest class size for which the chance that two or more students share a birthday exceeds one half?

$23$. The chance that $n$ birthdays are all different is $\prod_{k=0}^{n-1}\tfrac{365-k}{365}$. It first drops below $\tfrac12$ at $n=23$ (about $0.493$), so the collision probability is then about $0.507$.