1970 Canadian MO Problems/Problem 3

Problem

A set of balls is given. Each ball is coloured red or blue, and there is at least one of each colour. Each ball weighs either pound or pounds, and there is at least one of each weight. Prove that there are two balls having different weights and different colours.

Solution

Let each be the 4 types of balls such that the number is the weight and the letter be the color of the ball.

We can prove by contradiction.

Assume exists, then, can't exist. We know there has to be at least 1 of each color, so has to exist. Since exist, can't exist. The question states that there has to be at least 1 of each weight, but there isn't any ball that weighs 2 pounds. There is a contradiction.

Thus, there has to be a ball with different color and weight.