suppose that the vertices of a regular polygon of 20 sides are coloured with three colours-red,blue and green-such that there are exactly three red vertices. prove that there are three vertices A,B,C of the polygon having the same colour such that triangle ABC is isosceles.

There is only one possible patter in which the vertices are colored to get such a pattern where only three vertices are red, which is as follows: Color 1 2 times, Color 2 3 times and then color red( color 1 and 2 being blue or green.) We arrived at this pattern by the fact that 20/3= 6.6667, thus if red color is the sixth in the pattern, then only it will occur three times, and we also need a color to repeat itself 3 times. Now as we see in see pattern, there are 3 consecutive vertices colored with color 2. Let the first of these be A, B, C. Since the polygon is regular, therefore AB = BC. So triangle ABC is isosceles. Hope you understood the answer