I am stuck on this problem from a combinatorics book. I tried applying the pigeonhole principle to show that there are at least n/r numbers with the same color, and applied the formula for (x choose 2) to compute the number of pairs, but didn't get to the inequality we need to prove.
I also tried considering how many pairs are in each of the r pigeonholes, and considering that (x choose 2) is convex, I tried to apply Jensen's Inequality to simplify, but also did not reach the desired inequality. I would greatly appreciate any explanations with steps.