Given a row of $10^6$ lamps, all of them are initially off. Each of $2016$ people approaches the row and switches any $2017$ consecutive lamps. (One turns on lamps which are off, and turns off lamps which are on). It turned out that at the result $k$ lamps were on. How can I prove that among any $2017$ consecutive lamps there are not more than $\frac k2$ lamps which are turned on? It seems to me that it might be possible to prove this by induction, so I added the corresponding tag.
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I suspect that the specific numbers $10^6,2016$ and $2017$ are (almost) entirely arbitrary. It's important that $10^6\geq2017$, and it's important that $2016$ is even, but apart from that I think that you could use any numbers you want and it wouldn't affect the result. – Arthur Jan 21 '18 at 11:41