Problem

Source: Israel National Olympiad 2016 Q4

Tags: combinatorics, colorings, points



In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order: Choose two adjacent points with different colors, and add a point between them with one of the two colors only. Choose two adjacent points with the same color, and add a point between them with any of the three colors. Choose three adjacent points, at least two of them having the same color, and delete the middle point. Can Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?