BxMO 2025 Problem 2

Let $N \geq 2$ be a natural number. At a mathematical olympiad training camp, the same $N$ courses are organised every day. Each student takes exactly one of the $N$ courses each day. At the end of the camp, every student has taken each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of $N$ , the largest possible number of students at the camp?