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Prove that grasshopper can jump from any vertex to another

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December 31, 2011
combinatorics unsolvedcombinatorics

Problem Statement

Let CC be a circle, A1,A2,…,AnA_1 , A_2,\ldots ,A_n be distinct points inside CC and B1,B2,…,BnB_1 , B_2 ,\ldots ,B_n be distinct points on CC such that no two of the segments A1B1,A2B2,…,AnBnA_1B_1 , A_2 B_2 ,\ldots ,A_n B_n intersect. A grasshopper can jump from ArA_r to AsA_s if the line segment ArAsA_r A_s does not intersect any line segment AtBt(t≠r,s)A_t B_t (t \neq r, s). Prove that after a certain number of jumps, the grasshopper can jump from any AuA_u to any AvA_v .
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