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A_1,...,A_n points - Iran NMO 1999 (Second Round) Problem6

Source:

October 4, 2010

combinatorics proposedcombinatorics

Problem Statement

Let

A_1,A_2,\cdots,A_n

n

distinct points on the plane (

n>1

). We consider all the segments

A_iA_j

where

i<j\leq{n}

and color the midpoints of them. What's the minimum number of colored points? (In fact, if

k

colored points coincide, we count them

1