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 1
Using the logarithmic 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 .
Solution 2
Do as done in Solution 1 to get . Do as done in Solution 1 to get . If , then . If , then . Hence our final answer is -vsamc
See Also
2018 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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