2022 AMC 12A Problems/Problem 11
Contents
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 1
Let .
.
~ oinava
Solution 2
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 3 (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
Video Solution 1 (Understand the question first)
https://youtu.be/7yAh4MtJ8a8?si=CsVGUSiyUiT4nNI0&t=2076
~Math-X
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 |
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