2022 AIME I Problems/Problem 7
Problem
Let 
be distinct integers from 
to 
The minimum possible positive value of ![\[\dfrac{a \cdot b \cdot c - d \cdot e \cdot f}{g \cdot h \cdot i}\]](http://latex.artofproblemsolving.com/c/6/5/c653d108c03120ef617df93994006dd9a956c44c.png)
can be written as 
where 
and 
are relatively prime positive integers. Find 
Solution 1 (Optimization)
To minimize a positive fraction, we minimize its numerator and maximize its denominator. It is clear that 
If we minimize the numerator, then 
Note that 
so 
It follows that 
and 
are consecutive composites with prime factors no other than 
and 
The smallest values for and are 
and 
respectively. So, we have 
and 
from which 
If we do not minimize the numerator, then 
Note that 
Together, we conclude that the minimum possible positive value of 
is 
Therefore, the answer is 
~MRENTHUSIASM ~jgplay
Solution 2 (Bash)
Obviously, to find the correct answer, we need to get the largest denominator with the smallest numerator.
To bash efficiently, we can start out with 
as our denominator. This, however, leaves us with the numbers 
and 
left. The smallest we can make out of this is 
. When simplified, it gives us the answer of 
which gives a small answer of 
. Obviously there are larger answers than this.
After the first bash, we learn to bash even more efficiently, we can consider both the numerator and the denominator when guessing. We know the numerator has to be extremely small while still having a large denominator. When basing, we soon find out the couple 
and 
This gives us a numerator of 
which is by far the smallest yet. With the remaining numbers 
and 
, we get our answer of 
.
To finalize, we add up our numerator and denominator which gives us 
as our answer.
See Also
| 2022 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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