\(\hept{\begin{cases}a+b=c+d\Rightarrow\left(a+b\right)^2=\left(c+d\right)^2\Rightarrow a^2+2ab+b^2=c^2+2cd+d^2\\a^2+b^2=c^2+d^2\end{cases}}\)
\(\Rightarrow2ab=2cd\Rightarrow ab=cd\Rightarrow\frac{a}{d}=\frac{b}{c}=k\Rightarrow\hept{\begin{cases}a=dk\\b=ck\end{cases}}\)
Xét \(a^2+b^2=c^2+d^2\Leftrightarrow\left(dk\right)^2+b^2=\left(ck\right)^2+d^2\Leftrightarrow d^2\left(k^2-1\right)=b^2\left(k^2-1\right)\)
\(\Leftrightarrow\left(d^2-b^2\right)\left(k^2-1\right)=0\Leftrightarrow\orbr{\begin{cases}d^2-b^2=0\\k^2-1=0\end{cases}\Leftrightarrow\orbr{\begin{cases}d=\pm b\\k=\pm1\end{cases}}\Rightarrow\orbr{\begin{cases}a=\pm c\\a=\pm d;c=\pm b\end{cases}}}\)
\(\Rightarrow\orbr{\begin{cases}d^{2005}=b^{2005};a^{2005}=c^{2005}\\a^{2005}=d^{2005};c^{2005}=b^{2005}\end{cases}\Rightarrow\orbr{\begin{cases}a^{2005}+b^{2005}=c^{2005}+d^{2005}\\a^{2005}+b^{2005}=c^{2005}+d^{2005}\end{cases}}}\)
\(\Rightarrow a^{2005}+b^{2005}=c^{2005}+d^{2005}\left(đpcm\right)\)