1998 AIME Problems/Problem 14
An rectangular box has half the volume of an rectangular box, where and are integers, and What is the largest possible value of ?
Let’s solve for :
Clearly, we want to minimize the denominator, so we test . The possible pairs of factors of are . These give and respectively. Substituting into the numerator, we see that the first pair gives , while the second pair gives . We now check that is optimal, setting , in order to simplify calculations. Since We have Where we see gives us our maximum value of .
- Note that assumes , but this is clear as and similarly for .
Similarly as above, we solve for but we express the denominator differently:
Hence, it suffices to maximize under the conditions that is a positive integer.
Then since for we fix where we simply let to achieve
Observe that thus .
Now, we can use casework on and Simon's Favorite Factoring Trick to check that have no solution and for , we have the corresponding values of : .
Thus, the maximum value is .
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