I came across this interesting task. Fischer's inequality is well known and states the following:
Let $(X, B)$ be a $2$-design with parameters $(v, k, λ_2)$, where $v > k$. Then $|B| ≥ |X|$, which implies $b ≥ v$.
Our task is to characterize those $2$-designs where Fisher's inequality is achieved, i.e., $b = v$. So, $b = v$ if and only if ... (what)? Any idea how to begin?