# Difference between revisions of "Mock AIME I 2015 Problems/Problem 11"

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Let <math>\alpha = a</math>, <math>\beta = b</math>, and <math>\beta = c</math>. Then our system becomes | Let <math>\alpha = a</math>, <math>\beta = b</math>, and <math>\beta = c</math>. Then our system becomes | ||

− | <cmath>a + b + c</cmath> | + | <cmath>a + b + c = 6</cmath> |

<cmath>a^3 + b^3 + c^3 = 87</cmath> | <cmath>a^3 + b^3 + c^3 = 87</cmath> | ||

<cmath>(a + 1)(b + 1)(c + 1) = 33</cmath>. | <cmath>(a + 1)(b + 1)(c + 1) = 33</cmath>. |

## Revision as of 17:25, 11 October 2019

## 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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