Difference between revisions of "2017 AIME I Problems/Problem 10"
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==Problem 10== | ==Problem 10== | ||
| − | Let <math>z_1=18+83i,~z_2=18+39i,</math> and <math>z_3=78+99i,</math> where <math>i=\sqrt{-1}.</math> Let <math>z</math> be the unique complex number with the properties that <math>\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}</math> is a real number and the imaginary part of <math>z</math> is the greatest possible. Find the real part of <math>z</math> | + | Let <math>z_1=18+83i,~z_2=18+39i,</math> and <math>z_3=78+99i,</math> where <math>i=\sqrt{-1}.</math> Let <math>z</math> be the unique complex number with the properties that <math>\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}</math> is a real number and the imaginary part of <math>z</math> is the greatest possible. Find the real part of <math>z</math>. |
==Solution== | ==Solution== | ||
Revision as of 16:57, 8 March 2017
Problem 10
Let 
and 
where 
Let 
be the unique complex number with the properties that 
is a real number and the imaginary part of is the greatest possible. Find the real part of .
Solution
See Also
| 2017 AIME I (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 | ||
