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Symmetric Polynomials

In a symmetric polynomial, you can interchange any of the variables and get the same polynomial. Symmetric polynomials form the basis of Galois theory, which connects group theory and field theory.

Challenge Quizzes

Level 4

         

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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