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Algebra

# Symmetric Polynomials: Level 4 Challenges

Let $$a=2014^{2014}$$, $$b=2015^{2015}$$,$$c=2016^{2016}$$ . Find the value of $\frac{1}{2015^{a-a}+2015^{a-b}+2015^{a-c}} \\ +\frac{1}{2015^{b-a}+2015^{b-b}+2015^{b-c}} \\ +\frac{1}{2015^{c-a}+2015^{c-b}+2015^{c-c}}$

$\displaystyle \begin{cases} { a }^{ 1 }+{ b }^{ 1 }+{ c }^{ 1 }=\lambda \\ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }=\lambda \\ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }=\lambda \end{cases}$

If $$\lambda \in \mathbb{{Z}^{+}}$$ satisfying the system of equations above with $$abc=5!$$, determine $$\lambda$$.

Note: "$$!$$" represents factorial, not exclamatory sign.

$$x, y$$ and $$z$$ are complex numbers satisfying

$\begin{cases} x^1+y^1+z^1 & = 1\\ x^2 + y^2 + z^2 & = 2 \\ x^3 + y^3 + z^3 & = 3 \\ \end{cases}$

The value of $$x^4 + y^4 + z^4$$ can be expressed as $$\frac {a}{b}$$, where $$a$$ and $$b$$ are positive coprime integers. What is the value of $$a +b$$?

This problem is proposed by Harshit.

\large \begin{align*}\alpha+\beta+\gamma&=6 \\\alpha^3+\beta^3+\gamma^3&=87\\ (\alpha+1)(\beta+1)(\gamma+1)&=33 \end{align*}

Suppose $$\alpha$$, $$\beta$$, and $$\gamma$$ are complex numbers that satisfy the system of equations above.

If $$\frac1\alpha+\frac1\beta+\frac1\gamma=\tfrac mn$$ for positive coprime integers $$m$$ and $$n$$, find $$m+n$$.

If the roots of $$p(x) = x^3 + 3x^2 + 4x - 8$$ are $$\color{red}{a}$$, $$\color{blue}{b}$$ and $$\color{purple}{c}$$, what is the value of

$\color{red}{a}^2 \big(1 + \color{red}{a}^2\big) + \color{blue}{b}^2 \big(1 + \color{blue}{b}^2\big) + \color{purple}{c}^2 \big(1 + \color{purple}{c}^2\big)?$

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