2009 AIME I Problems/Problem 14
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Problem
For 
, define 
, where 
. If 
and 
, find the minimum possible value for 
.
Solution
Because the order of the 
s doesn't matter, we simply need to find the number of 
s 
s 
s and 
s that minimize . So let 
and 
represent the number of s, s, s, and s respectively. Then we can write three equations based on these variables. Since there are a total of 
s, we know that 
. We also know that 
and 
. We can now solve these down to two variables:
Substituting this into the second and third equations, we get
and
The second of these can be reduced to
Now we substitute 
from the first new equation into the other new equation.
Since 
and are integers, the two solutions to this are 
or 
.
If you plug both these solutions in to it is apparent that the second one returns a smaller value. It turns out that 
, 
, 
, and 
, so 
.
See also
| 2009 AIME I (Problems • Resources) | ||
| Preceded by Problem 13 | Followed by Problem 15 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
