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bug traveling in the circle

Source: Russia 1994

October 7, 2008
combinatorics unsolvedcombinatorics

Problem Statement

Points A1,A2,...,An A_1,A_2, ... ,A_n inside a circle and points B1,B2,...,Bn B_1,B_2,...,B_n on its boundary are positioned so that the segments A1B1,A2B2,...,AnBn A_1B_1,A_2B_2, ... ,A_nB_n do not intersect. A bug can go from point Ai A_i to Aj A_j if the segment AiAj A_iA_j does not intersect any segment AkBk A_kB_k, k≠i,j k \neq i, j. Prove that the bug can go from any point Ap A_p to any point Aq A_q in a finite number of steps.
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