Mock AIME I 2015 Problems/Problem 11
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, .
<baker77>