Question 135
Let $A_1A_2\dots A_n$ be a convex polygon where $n \ge 4$. For each $i = 1, 2, \dots, n$, let $f(i)$ be the maximum of the angles $\angle A_iA_jA_{i+1}$ where $A_j$ can be any vertex different from $A_i$ and $A_{i+1}$ (with $A_{n+1} = A_1$). For each $n$, find the smallest possible value of $f(1) + f(2) + \dots + f(n)$.