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1991 APMO Problems/Problem 2: Difference between revisions

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== Problem ==
== Problem ==


There are <math>997</math> points in the plane. Show that they have at least <math>1991</math> distinct midpoints. Is it possible to have exactly <math>1991</math> midpoints?
Suppose there are <imath>997</imath> 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 <imath>1991</imath> red points in the plane. Can you find a special case with exactly <imath>1991</imath> red points?


== See Also ==
== See Also ==

Revision as of 00:48, 9 November 2025

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?

See Also

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