Special students

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Source: 2013 Argentina L2 P4

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#1 h Dec 24, 2024, 5:47 PM

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In a school with double schooling, in the morning the language teacher divided the students into $200$ groups for an activity. In the afternoon, the math teacher divided the same students into $300$ groups for another activity. A student is considered special if the group they belonged to in the afternoon is smaller than the group they belonged to in the morning. Find the minimum number of special students that can exist in the school.

Note: Each group has at least one student.

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#2 h Mar 30, 2025, 8:04 PM

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The answer is $101$ .
For the construction, let one group in the morning have $101$ , and let the other $199$ groups have a random grouping. Let then the $101$ students be distributed into $101$ distinct groups in the afternoon, and let the $199$ groups remain the same as in the morning. Its easy to see that there are $101$ special students.

For the bound, assign a value to each student in both the morning and the afternoon. Assign it like this: If a group has $k$ students, let each of these $k$ students have value $\frac{1}{k}$ . The sum of values in the morning is $200$ , while in the afternoon the sum of values if $300$ . If a student isnt special, that students value doesnt increase, so only special students contribute to increasing the value. Note also that a single student cannot increase its value by more than or equal to $1$ , but as the difference in value in the moring and afternoon is $100$ . there has to be more than $100$ special students, as desired.

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