1983 IMO 第 1 题

(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.

(AUS 1) $P_{1}、P_{2}、\ldots、P_{1983}$ 地区由 10 家国际航空公司 $A_{1}、A_{2}、\ldots、A_{10}$ 提供服务。请注意，这些地点中的任何两个地点之间都有直达航班(无停靠)，并且所有航空公司的航班时刻表均提供往返航班。证明至少有一家航空公司可以提供单次着陆的往返旅行。