Algebra
# Symmetric Polynomials

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

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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