1991 AJHSME Problems/Problem 17
Problem
An auditorium with rows of seats has seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students that can be seated for an exam is
Solution
We first note that if a row has seats, then the maximum number of students that can be seated in that row is , where is the smallest integer greater than or equal to . If a row has seats, clearly we can only fit students in that row. If a row has seats, we can fit students by putting students at the ends and then alternating between skipping a seat and putting a student in.
For each row with seats, there is a corresponding row with seats. The sum of the maximum number of students for these rows is There are pairs of rows, so the maximum number of students for the exam is .
See Also
1991 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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