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MMO 207 Moscow MO 1951 bus route, 14 stops, 25 passengers

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August 7, 2019
combinatorics

Problem Statement

* A bus route has 1414 stops (counting the first and the last). A bus cannot carry more than 2525 passengers. We assume that a passenger takes a bus from AA to BB if (s)he enters the bus at AA and gets off at BB. Prove that for any bus route:
a) there are 88 distinct stops A1,B1,A2,B2,A3,B3,A4,B4A_1, B_1, A_2, B_2, A_3, B_3, A_4, B_4 such that no passenger rides from AkA_k to BkB_k for all k=1,2,3,4k = 1, 2, 3, 4 (#)
b) there might not exist 1010 distinct stops A1,B1,...,A5,B5A_1, B_1, . . . , A_5, B_5 with property (#).
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