2022 CMO 第 4 题

A conference is attended by $n (n\ge 3)$ scientists. Each scientist has some friends in this conference (friendship is mutual and no one is a friend of him/herself). Suppose that no matter how we partition the scientists into two nonempty groups, there always exist two scientists in the same group who are friends, and there always exist two scientists in different groups who are friends.

A proposal is introduced on the first day of the conference. Each of the scientists' opinion on the proposal can be expressed as a non-negative integer. Everyday from the second day onwards, each scientists' opinion is changed to the integer part of the average of his/her friends' opinions from the previous day.

Prove that after a period of time, all scientists have the same opinion on the proposal.

$n (n\ge 3)$ 名科学家参加了一次会议。每个科学家在这个会议上都有一些朋友(友谊是相互的，没有人是他/她自己的朋友)。假设无论我们如何将科学家分为两个非空组，同一组中总是存在两个作为朋友的科学家，并且不同组中总是存在两个作为朋友的科学家。

会议第一天提出了一项提案。每个科学家对该提案的意见都可以表示为一个非负整数。从第二天开始，每位科学家的意见都会更改为前一天他/她的朋友意见平均值的整数部分。

证明一段时间后，所有科学家对该提案的看法一致。