Problem:
At the beginning of the school year, of all students in Mr. Wells' math class answered "Yes" to the question "Do you love math", and answered "No." At the end of the school year, answered "Yes" and answered "No." Altogether, of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of ?
Answer Choices:
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Solution:
Assume there are students in Mr. Wells' class. Then at least students answered "No" at the beginning of the school year and "Yes" at the end, so . Because only students answered "No" at the end of the school year, at least students who answered "Yes" at the beginning of the year gave the same answer at the end, so . The difference between the maximum and minimum possible values of is . The minimum is achieved if exactly students answered "No" at the beginning and "Yes" at the end of the school year. The maximum is achieved if exactly students answered "Yes" at the beginning and the end.
The problems on this page are the property of the MAA's American Mathematics Competitions