Question 1
(AUS 1) The localities $P_1,P_2,\ldots,P_{1983}$ are served by ten international airlines $A_1,A_2,\ldots,A_{10}$. It is noticed that there is direct service (without stops) between any two of these localities 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.