2018 AIME I Problems/Problem 5
Contents
Problem 5
For each ordered pair of real numbers satisfying
there is a real number
such that
Find the product of all possible values of
.
Solution
Using the log property , we note that
.
That gives
upon simplification and division by
. Then,
or
.
From the second equation,
. If we take
, we see that
. If we take
, we see that
. The product is
.
-expiLnCalc
Note
The cases and
can be found by SFFT (Simon's Favorite Factoring Trick) from
.
See Also
2018 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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