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Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.
If \(P(n)=\frac{n^3+n^2-2n}{n^2-4}+\frac{n+1}{2+n-n^2}\), what is the value of \(\lfloor P(409) \rfloor + \lfloor P(423) \rfloor\)?
Details and assumptions
The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example, \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).
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If \(x\) and \(y\) are real numbers such that \[x\neq y, x+y=5, \mbox{ and } xy=2,\] what is the value of \[\frac{x^8-y^8}{433(x-y)}?\]
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If \(\frac{x^2+y^2+z^2}{xy+yz+zx}=7\) and \(\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}=6\), what is the value of \(\frac{(x+y+z)^3}{xyz}\)?
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Let \(a\) be a positive number such that \[a^2+\frac{1}{a^2}=3.\] If the value of \[a^3+\frac{1}{a^3}\] can be expressed as \(m\sqrt{n},\) where \(n\) is a prime number, what is \(m+n?\)
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\(x, y\) and \(z\) are complex numbers such that
\[ \begin{cases} x + y + z & = 46 \\ (x-y)^2 + (y-z)^2 + (z-x) ^ 2 & = xyz.\\ \end{cases} \]
What is the value of \( \frac{ x^3 + y^3 + z^3} { xyz} \)?
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