2024 AMC 12B Problems/Problem 13
Contents
[hide]Problem 13
There are real numbers and that satisfy the system of equationsWhat is the minimum possible value of ?
Solution 1 (Easy and Fast)
Adding up the first and second equation, we get: All squared values must be greater than or equal to . As we are aiming for the minimum value, we set the two squared terms to be .
This leads to
~mitsuihisashi14
Solution 2 (Coordinate Geometry and QM-GM Inequality)
Distance between 2 circle centers is the 2 circle must intersect given there exists one or more pair of (x,y), connecting and any one of the 2 circle intersection point we get a triangle with 3 sides ( radius () , radius () , ) , then the equal sign will be reached when 2 circles are external tangent to each other,
Apply QM-GM inequality in step below, we get
Therefore, h + k .
Solution 3
~Kathan
See also
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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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