IMO 1983 LL AUS2

Seventeen cities are served by four airlines. It is noted that

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IMO 1983 LL AUS2

Origin: AUS

Problem

Seventeen cities are served by four airlines. It is noted that there is direct service (without stops) between any two cities and that all airline schedules offer round-trip flights. Prove that at least one of the airlines can offer a round trip with an odd number of landings.