Difference between revisions of "2021 AIME II Problems/Problem 4"

(Problem)
(Solution)
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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==
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==Solution 1==
We can't have a solution without a problem.
+
 
 +
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 23:37, 22 March 2021

Problem

There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \sqrt{n} \cdot i,$ where $m$ and $n$ are positive integers and $i = \sqrt{-1}.$ Find $m+n.$

Solution 1

Conjugate root theorem

Solution in progress

~JimY

Solution 2

See also

2021 AIME II (ProblemsAnswer KeyResources)
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

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