Question 31
Let $A_1A_2\dots A_{99}$ be a regular polygon and $A_{100}$ its center. Ali wants to draw each of the segments $A_iA_j$ ($1 \le i \lt j \le 100$) using one of $k$ colors such that no two segments of the same color share an endpoint or cross. Find the smallest positive integer $k$ for which such an operation is possible.