On the floor of a room of area 5, you place 9 rugs. Each is an arbitrary shape but has area 1. Prove that there are two rugs that overlap by at least 1/9.
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Suppose that this isn’t true — that every pair of rugs overlaps by less than 1/9. Place the rugs on the floor one by one and note how much of the bare floor each succeeding rug must cover. The first rug must cover area 1, or 9/9. The second must cover an area greater than 8/9 (because less than 1/9 can overlap the first rug). The third rug can overlap the first two somewhat, but must still cover an area greater than 7/9 of the bare floor. And so on: The fourth, fifth, … ninth rug will cover an area greater than 6/9, 5/9, … 1/9 on the bare floor. Since 9/9 + … + 1/9 = 5, the nine rugs together must cover an area greater than 5. That’s a contradiction, so our supposition can’t be true — some pair of rugs must overlap by at least 1/9.
From Arthur Engel, Problem-Solving Strategies, 1998.
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