Symmetric Polynomials
Let \(a=2014^{2014}\), \(b=2015^{2015}\),\(c=2016^{2016}\) . Find the value of \[ \frac{1}{2015^{a-a}+2015^{a-b}+2015^{a-c}} \\ +\frac{1}{2015^{b-a}+2015^{b-b}+2015^{b-c}} \\ +\frac{1}{2015^{c-a}+2015^{c-b}+2015^{c-c}} \]
by
Hoo Zhi Yee
\[\displaystyle \begin{cases} { a }^{ 1 }+{ b }^{ 1 }+{ c }^{ 1 }=\lambda \\ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }=\lambda \\ { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 }=\lambda \end{cases}\]
If \(\lambda \in \mathbb{{Z}^{+}}\) satisfying the system of equations above with \(abc=5!\), determine \(\lambda\).
Note: "\(!\)" represents factorial, not exclamatory sign.
by
Dylan Pentland
\(x, y\) 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.
by
Calvin Lin
\[\large \begin{align*}\alpha+\beta+\gamma&=6 \\\alpha^3+\beta^3+\gamma^3&=87\\ (\alpha+1)(\beta+1)(\gamma+1)&=33 \end{align*}\]
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 \(m+n\).
by
David Altizio
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
\[\color{red}{a}^2 \big(1 + \color{red}{a}^2\big) + \color{blue}{b}^2 \big(1 + \color{blue}{b}^2\big) + \color{purple}{c}^2 \big(1 + \color{purple}{c}^2\big)?\]
by
Sharky Kesa