Solution
= Solution
For <matching one specified birthday>, each of the $N-1$ other guests avoids the host's date with probability $364/365$. <Independence> gives
$$
\boxed{P(\text{at least one matches the host})=1-\left(\frac{364}{365}\right)^{N-1}.}
$$
For this to exceed $1/2$, the negative logarithm of $364/365$ reverses the inequality on division, giving $N-1>\ln(1/2)/\ln(364/365)\simeq252.652$. Thus \b[the smallest total attendance is $N=254$], including the host.