Difference between revisions of "2022 AMC 12A Problems/Problem 11"
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− | Let <math>a = 2 \cdot |\log_6 10 - 1| = |\log_6 9 - \log_6 x| </math>. | + | Let <math>a = 2 \cdot |\log_6 10 - 1| = |\log_6 9 - \log_6 x| = |\log_6 \frac{9}{x}| </math>. |
− | <math> | + | <math> \pm a = \log_6 \frac{9}{x} x \implies 6^{\pm a} = b^{\pm 1} = 9/x \implies x = 9 \cdot b^{\pm 1} </math> |
− | <math> ( | + | <math> (9b^1) \cdot (9b^{-1}) = 9 \cdot 9 = \boxed{81}</math>. |
~ oinava | ~ oinava |
Revision as of 18:08, 13 March 2023
Contents
[hide]Problem
What is the product of all real numbers such that the distance on the number line between and is twice the distance on the number line between and ?
Solution
Let .
.
~ oinava
Solution
First, notice that there must be two such numbers: one greater than and one less than it. Furthermore, they both have to be the same distance away, namely . Let these two numbers be and . Because they are equidistant from , we have . Using log properties, this simplifies to . We then have , so .
~ jamesl123456
Solution 2 (Logarithmic Rules and Casework)
In effect we must find all such that where .
Notice that by log rules Using log rules again,
Now we proceed by casework for the distinct values of .
Case 1
Subbing in for and using log rules, From this we may conclude that
Case 2
Subbing in for and using log rules, From this we conclude that
Finding the product of the distinct values,
~Spektrum
Video Solution 1 (Quick and Simple)
~Education, the Study of Everything
See Also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.