题面据 IMO Shortlist 可核档案整理；中文题意为本站自译，公式请以原始来源为准。
In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible number of flights required to go from one of them to the other. It is known that for any city there are at most 100 cities at distance exactly three from it. Prove that there is no city such that more than 2550 other cities have distance exactly four from it. (Russia)
在某些国家/地区，几对城市之间有双向直飞航班。可以通过一系列航班从任何城市前往任何其他城市。两个城市之间的距离定义为从其中一个城市飞往另一个城市所需的最少航班数量。众所周知，对于任何一个城市来说，最多有100个距离它正好三倍的城市。证明不存在一个城市可以让超过 2550 个其他城市与它的距离恰好为 4。 (俄罗斯)
提示 1
先决定对象是什么：集合、图、排列、颜色、路径，还是一次操作后的状态。
提示 2
找一个极端对象、双计数式、不变量，或把限制转成图上的局部条件。
提示 3
把局部限制累加成全局矛盾，或给出覆盖全部情形的构造。
完整解答
这页先给题面、题型和提示阶梯，完整证明留给读者逐步展开。2013 年 IMO Shortlist C6 可先归入组合：第一步把题设翻成对象、条件、目标三行；第二步沿提示寻找不变量、标准构型或关键变形；第三步补齐边界情形，并回到题目原要求核对。
这题适合先独立想一轮再打开提示。不要急着搜索完整解答，先问自己：题面里最硬的限制是哪一句？