Problem: Any five points are taken inside or on a square of side . Let be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than . Then is:
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Solution:
The greatest distance between two points and is the length of a diagonal, . Place at the four corners. For the fifth point to be as far as possible from the other four points, it must be located at the center of the square. The distance from to any of the other four points is .
Location of the points inside the square will yield distances between pairs of points, the smallest of which is less than .