1991 APMO Problems/Problem 2: Difference between revisions
Created page with "There are 997 points in the plane. Show that they have at least 1991 distinct midpoints. Is it possible to have exactly 1991 midpoints?" |
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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? | |||
== 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 <imath>(0, 0), (0, 1), \ldots, (0, 996)</imath>. Notice that possible red points will be <imath>(0, 0.5), (0, 1), (0, 1.5), \ldots, (0, 995.5)</imath>. Counting, there are <imath>\frac{995.5}{0.5} = 1991</imath> red dots. | |||
~MaPhyCom | |||
== See Also == | |||
* [[APMO]] | |||
* [[APMO Problems and Solutions]] | |||
* [[1991 APMO Problems]] | |||
Latest revision as of 01:21, 9 November 2025
Suppose there are 
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 
red points in the plane. Can you find a special case with exactly 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 
. Notice that possible red points will be 
. Counting, there are 
red dots.
~MaPhyCom
