2001 AIME II Problems/Problem 2
Each of the students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between percent and percent of the school population, and the number who study French is between percent and percent. Let be the smallest number of students who could study both languages, and let be the largest number of students who could study both languages. Find .
Let be the percent of people who study Spanish, be the number of people who study French, and let be the number of students who study both. Then , and . By the Principle of Inclusion-Exclusion,
For to be smallest, and must be minimized.
For to be largest, and must be maximized.
Therefore, the answer is .
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