Problem:
A drawer contains a mixture of red socks and blue socks, at most in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
Solution:
Let and , respectively, denote the numbers of red and blue socks in the drawer. Because the probability of obtaining a non-matching pair is , we have
This leads to , which can be written as . This shows that the total number of socks in the drawer is a perfect square. Let , so . Then . Since , we must have . We then see that the largest possible value of occurs when , and this value of is .
The problems on this page are the property of the MAA's American Mathematics Competitions