1998 AIME Problems/Problem 14
Problem
An rectangular box has half the volume of an
rectangular box, where
and
are integers, and
What is the largest possible value of
?
Solution
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
.
See also
1998 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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