Difference between revisions of "2012 AIME II Problems/Problem 8"
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== Problem 8 == | == Problem 8 == | ||
− | The complex numbers <math>z</math> and <math>w</math> satisfy the system <cmath> z + \frac{20i}w = 5+i | + | The complex numbers <math>z</math> and <math>w</math> satisfy the system <cmath> z + \frac{20i}w = 5+i </cmath> |
− | w+\frac{12i}z = -4+10i </cmath> Find the smallest possible value of <math>\vert zw\vert^2</math>. | + | <cmath> w+\frac{12i}z = -4+10i </cmath> Find the smallest possible value of <math>\vert zw\vert^2</math>. |
== Solution == | == Solution == |
Revision as of 21:00, 8 March 2015
Problem 8
The complex numbers and satisfy the system Find the smallest possible value of .
Solution
Multiplying the two equations together gives us and multiplying by then gives us a quadratic in : Using the quadratic formula, we find the two possible values of to be = The smallest possible value of is then obviously
See Also
2012 AIME II (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 | ||
All AIME Problems and Solutions |
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