Problem

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Tags: combinatorics unsolved, combinatorics



Let $C$ be a circle, $A_1 , A_2,\ldots ,A_n$ be distinct points inside $C$ and $B_1 , B_2 ,\ldots ,B_n$ be distinct points on $C$ such that no two of the segments $A_1B_1 , A_2 B_2 ,\ldots ,A_n B_n$ intersect. A grasshopper can jump from $A_r$ to $A_s$ if the line segment $A_r A_s$ does not intersect any line segment $A_t B_t (t \neq r, s)$. Prove that after a certain number of jumps, the grasshopper can jump from any $A_u$ to any $A_v$ .