Advanced factorization is a gateway to algebraic number theory, which mathematicians study in order to solve famous conjectures like Fermat's Last Theorem.

Given that
\[\begin{align}
& x = -a+b+c, \\
& y = a-b+c, \\
& z = a+b-c ,\\
& a^2+b^2+c^2 =20,
\end{align}\]
find the value of \(x^2+y^2+z^2+xy+yz+zx\).

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Suppose that \(a\) is an integer such that \[x^2+2xy-80y^2+ax+18y+81\] can be factorized as the product of linear expressions in \(x\) and \(y\). What is the value of \(a\)?

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Suppose \(a\), \(b\) and \(c\) are real numbers such that \[a+b+c=5.\] If \[x=a-2b+3c, y=b-2c+3a, z=c-2a+3b,\] what is the value of \[(x^2+2xy+4)+(y^2+2yz+4)+(z^2+2zx+4)?\]

Excel in math and science

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