Problem:
There are people in a room. What is the largest value of such that the statement "At least people in this room have birthdays falling in the same month." is always true?
Answer Choices:
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Solution:
If no more than people have birthdays in any month, then at most people would be accounted for. Therefore the statement is true for . The statement is false for if, for example, people have birthdays in each of the first months of the year, and people have birthdays in each of the last months, for a total of people.
The average number of birthdays per month is , which is strictly between and . Therefore at least one month must contain at least birthdays, and, as above, it is possible to distribute the birthdays so that all months contain or birthdays.
The problems on this page are the property of the MAA's American Mathematics Competitions