Can the vertices of a regular $20$-gon be labeled with numbers $1, 2, ..., 20$ in such a way that each label is used exactly once and for every four consecutive vertices the sum of their labels is less than $43$?
First I tried to find some example. Then I got nothing , so decided to try something else. I wrote that sum of $1+2+3...20=210<43×5$ So it seems to have a solution. But I think something is disturbing this problem to have a solution. I tried to group all numbers into 4 parts . If I colour all verexes in 4 colours it will be 1,2,3,4,1,2,3,4....4 on 20-gon so if I choose 4 consicutive vertexes they all will be different colours. That means that there's a way to divide all 20 numbers into 4 groups of each colour or there's not , so we've to prove that it's impossible.