2005 AIME I Problems/Problem 4
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Problem
The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 
members left over. The director realizes that if he arranges the group in a formation with 
more rows than columns, there are no members left over. Find the maximum number of members this band can have.
Contents |
Solution
Solution 1
If 
then 
and so 
. If 
is an integer there are no numbers which are 5 more than a perfect square strictly between 
and 
. Thus, if the number of columns is , the number of students is 
which must be 5 more than a perfect square, so 
. In fact, when 
we have 
, so this number works and no larger number can. Thus, the answer is 
.
Solution 2
Define the number of rows/columns of the square formation as 
, and the number of rows of the rectangular formation 
(so there are 
columns). Thus, 
. The quadratic formula yields 
. 
must be an integer, say 
. Then 
and 
. The factors of 
are 
; is maximized for the first case. Thus, 
, and 
. The latter obviously can be discarded, so there are 
rows and 
columns, making the answer 
.
Solution 3
The number of members is 
for some and 
. Multiply both sides by 
and complete the square to get 
. Thus, we have 
. Since we want to maximize , set the first factor equal to and the second equal to 
. Solving gives 
, so the answer is 
.
See also
| 2005 AIME I (Problems • Resources) | ||
| Preceded by Problem 3 | Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
