Difference between revisions of "2019 AIME I Problems/Problem 12"
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==Problem 12== | ==Problem 12== | ||
| − | + | Given <math>f(z) = z^2-19z</math>, there are complex numbers <math>z</math> with the property that <math>z</math>, <math>f(z)</math>, and <math>f(f(z))</math> are the vertices of a right triangle in the complex plane with a right angle at <math>f(z)</math>. There are positive integers <math>m</math> and <math>n</math> such that one such value of <math>z</math> is <math>m+\sqrt{n}+11i</math>. Find <math>m+n</math>. | |
==Solution== | ==Solution== | ||
Revision as of 22:18, 14 March 2019
The 2019 AIME I takes place on March 13, 2019.
Problem 12
Given 
, there are complex numbers 
with the property that , 
, and 
are the vertices of a right triangle in the complex plane with a right angle at . There are positive integers 
and 
such that one such value of is 
. Find 
.
Solution
See Also
| 2019 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
