An country consists of 100 equal square districts, laid out in a 10x10 grid, with an island in the center of each of the 100 districts. The country wants to better connect the islands, building bridges in a straight line directly from one island to another. Consider five different islands. Suppose bridges are built between each possible pair of islands within these five. Prove that at least one of these bridges will be built directly over some island in the very middle of its path. Note that this "midpoint island" does not have to be one of the five islands that are having bridges buil
An country consists of 100 equal square districts, laid out in a 10x10 grid, with an island in the center of each of the 100 districts.
The country wants to better connect the islands, building bridges in a straight line directly from one island to another.
Consider five different islands. Suppose bridges are built between each possible pair of islands within these five.
Prove that at least one of these bridges will be built directly over some island in the very middle of its path. Note that this "midpoint island" does not have to be one of the five islands that are having bridges built.
Let us consider coordinates of each of the island as where are integers between
Since the integers can be both even and odd So, we have four types of islands
We have to take five islands, and there are only four types of possible islands. So, by pigeonhole principle, At least two islands are of the same type.
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