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。
A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off.
*Proposed by Australia*
一座房子的房间里分布着偶数盏灯，每个房间至少有三盏灯。每盏灯与另一盏灯共用一个开关，不一定来自同一个房间。两个灯共用的开关的每次变化都会同时改变它们的状态。证明对于灯的每个初始状态，某些开关中存在一系列变化，在这些开关的末端，每个房间都包含打开的灯和关闭的灯。
*由澳大利亚提出*
提示 1
先猜等号形状，再看同次性、归一化和每一项的量纲。
提示 2
试着把式子拆成均值、柯西、凸性、重排或切线法可处理的块。
提示 3
最后检查等号条件和边界情形是否都与题设兼容。
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。2005 年 IMO Shortlist S02 可先归入不等式：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？