2013 AMC 12A Problems/Problem 8
Contents
Problem
Given that and are distinct nonzero real numbers such that , what is ?
Solution 1
Since , we may assume that and/or, equivalently, .
Cross multiply in either equation, giving us .
Solution 2
Since
Solution 3
Let Consider the equation Reorganizing, we see that satisfies Notice that there can be at most two distinct values of which satisfy this equation, and and are two distinct possible values for Therefore, and are roots of this quadratic, and by Vieta’s formulas we see that thereby must equal
~ Professor-Mom
Solution 4
Multiply both sides by xy to get
Rearrange to get
Factor out on the left side and on the right side to get
Divide by {You can do this since x and y are distinct} to get
Video Solution
https://youtu.be/ba6w1OhXqOQ?t=1129
~ pi_is_3.14
Video Solution
https://youtu.be/CCjcMVtkVaQ ~sugar_rush
See also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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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