1991 AJHSME Problems/Problem 17
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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 • Resources) | ||
| Preceded by Problem 16 | Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
