Difference between revisions of "2021 AIME II Problems/Problem 4"
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There are real numbers <math>a, b, c,</math> and <math>d</math> such that <math>-20</math> is a root of <math>x^3 + ax + b</math> and <math>-21</math> is a root of <math>x^3 + cx^2 + d.</math> These two polynomials share a complex root <math>m + \sqrt{n} \cdot i,</math> where <math>m</math> and <math>n</math> are positive integers and <math>i = \sqrt{-1}.</math> Find <math>m+n.</math> | There are real numbers <math>a, b, c,</math> and <math>d</math> such that <math>-20</math> is a root of <math>x^3 + ax + b</math> and <math>-21</math> is a root of <math>x^3 + cx^2 + d.</math> These two polynomials share a complex root <math>m + \sqrt{n} \cdot i,</math> where <math>m</math> and <math>n</math> are positive integers and <math>i = \sqrt{-1}.</math> Find <math>m+n.</math> | ||
| − | ==Solution== | + | ==Solution 1== |
| − | + | ||
| + | Conjugate root theorem | ||
| + | |||
| + | Solution in progress | ||
| + | |||
| + | ~JimY | ||
| + | |||
| + | ==Solution 2== | ||
==See also== | ==See also== | ||
{{AIME box|year=2021|n=II|num-b=3|num-a=5}} | {{AIME box|year=2021|n=II|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 22:37, 22 March 2021
Contents
Problem
There are real numbers 
and 
such that 
is a root of 
and 
is a root of 
These two polynomials share a complex root 
where 
and 
are positive integers and 
Find 
Solution 1
Conjugate root theorem
Solution in progress
~JimY
Solution 2
See also
| 2021 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 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. 
