Difference between revisions of "2018 AIME I Problems/Problem 5"
(→Solution 2) |
|||
Line 17: | Line 17: | ||
-vsamc<math>\newline</math> | -vsamc<math>\newline</math> | ||
-minor edit:einsteinstudent | -minor edit:einsteinstudent | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://www.youtube.com/watch?v=iE8paW_ICxw | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2018|n=I|num-b=4|num-a=6}} | {{AIME box|year=2018|n=I|num-b=4|num-a=6}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:31, 23 July 2020
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
-minor edit:einsteinstudent
Video Solution
https://www.youtube.com/watch?v=iE8paW_ICxw
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.