A group of $n$ friends takes $r$ distinct photos (two photos do not have exactly the same people) which contain at least one photo. Find the largest $r$ such that for each pair of photos, there is at least one person who appears in both.
We show that $r \leqslant 2^{n-1}$. For each photo taken, we cannot take its complement (the photo with all the people who are not in the photo we take) because there will be no person in common between the two photos. In total, there are $2^{n}$ possible photos with $n$ people. Therefore, we can take a maximum of $2^{n} / 2=2^{n-1}$ photos. Construction: We can keep one person in all the photos and choose whether the $n-1$ others are in the photo. This gives us $2^{n-1}$ photos.