Difference between revisions of "Mock AIME I 2015 Problems/Problem 11"
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==Problem== | ==Problem== | ||
− | Suppose <math>\alpha</math>, <math>\beta</math>, and <math>\gamma</math> are complex numbers that satisfy the system of equations | + | Suppose <math>\alpha</math>, <math>\beta</math>, and <math>\gamma</math> are complex numbers that satisfy the system of equations <math>\begin{align*}\alpha+\beta+\gamma&=6,\\alpha^3+\beta^3+\gamma^3&=87,\(\alpha+1)(\beta+1)(\gamma+1)&=33</math>.\end{align*}If <math>\frac1\alpha+\frac1\beta+\frac1\gamma=\tfrac mn</math> for positive relatively prime integers <math>m</math> and <math>n</math>, find <math>m+n</math>. |
==Solution 1== | ==Solution 1== |
Revision as of 10:25, 29 October 2019
Problem
Suppose , , and are complex numbers that satisfy the system of equations $\begin{align*}\alpha+\beta+\gamma&=6,\\alpha^3+\beta^3+\gamma^3&=87,\(\alpha+1)(\beta+1)(\gamma+1)&=33$ (Error compiling LaTeX. Unknown error_msg).\end{align*}If for positive relatively prime integers and , find .
Solution 1
For convenience, let's use instead of . Define a polynomial such that . Let and . Then, our polynomial becomes . Note that we want to compute .
From the given information, we know that the coefficient of the term is , and we also know that , or in other words, . By Newton's Sums (since we are given ), we also find that . Solving this system, we find that . Thus, , so our final answer is .
Solution 2
Let , , and . Then our system becomes .
Since , this equation becomes .
. Since , this equation becomes .
We will now use these equations to solve the problem. Let , and . Then we have . Our solutions are and .
Then . So, .
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