1991 APMO Problems/Problem 2

Problem

Suppose there are $997$ points given in a plane. If every two points are joined by a line segment with its midpoint colored in red, show that there are at least $1991$ red points in the plane. Can you find a special case with exactly $1991$ red points?

Solution

For the number of red points to be minimum, the number of red points that coincide must be maximum. Therefore, set all points as $(0, 0), (0, 1), \ldots, (0, 996)$ . Notice that possible red points will be $(0, 0.5), (0, 1), (0, 1.5), \ldots, (0, 995.5)$ . Counting, there are $\frac{995.5}{0.5} = 1991$ red dots.

~MaPhyCom

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1991 APMO Problems/Problem 2

Problem

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