# Mock AIME I 2015 Problems/Problem 11

## Problem

Suppose , , and are complex numbers that satisfy the system of equations 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 .

Therefore, . So, .

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