\(x^2+y^2=\left(x+y\right)^2-2xy=25-2xy\)
\(x^4+y^4=\left(x^2+y^2\right)^2-2x^2y^2=\left(25-2xy\right)^2-2x^2y^2\)
\(\Rightarrow x^4+y^4=2x^2y^2-100xy+625\)
Thay vào ta được:
\(2x^2y^2-100xy+625+25-2xy=110\)
\(\Leftrightarrow2x^2y^2-102xy+540=0\Rightarrow\left[{}\begin{matrix}xy=45>\frac{\left(x+y\right)^2}{4}\left(l\right)\\xy=5\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x+y=5\\xy=6\end{matrix}\right.\) \(\Rightarrow\left(x;y\right)=\left(2;3\right);\left(3;2\right)\)
\(\Rightarrow P=2^5+3^5=275\)
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