Socialists Republic Of Czechoslovakia 9
Source: IMO LongList 1959-1966 Problem 43
September 2, 2004
combinatoricsColoringgraph theoryRamsey TheoryExtremal combinatoricsIMO LonglistIMO Shortlist
Problem Statement
Given points in a plane, no three of them being collinear. Each two of these points are joined with a segment, and every of these segments is painted either red or blue; assume that there is no triangle whose sides are segments of equal color.
a.) Show that:
(1) Among the four segments originating at any of the points, two are red and two are blue.
(2) The red segments form a closed way passing through all given points. (Similarly for the blue segments.)
b.) Give a plan how to paint the segments either red or blue in order to have the condition (no triangle with equally colored sides) satisfied.