Question

[dbrby]

cho a,b,c,d thỏa mãn a+b=c+d và \(a^2+b^2=c^2+d^2\)

Cmr \(a^{2012}+b^{2012}=c^{2012}+d^{2012}\)

Answer 1

[Khôi Bùi]

Ta có : \(a^2+b^2=c^2+d^2\)

\(\Leftrightarrow a^2-c^2=d^2-b^2\)

\(\Leftrightarrow\left(a-c\right)\left(a+c\right)=\left(d-b\right)\left(d+b\right)\)

Do \(a+b=c+d\Rightarrow a-c=d-b\)

\(\Rightarrow\left(a-c\right)\left(a+c\right)=\left(a-c\right)\left(d+b\right)\)

\(\Leftrightarrow\left(a-c\right)\left(a+c-b-d\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}a-c=0=d-b\\a+c=b+d\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}a=c\\d=b\end{matrix}\right.\\a+c=b+d\end{matrix}\right.\)

Với a = c ; d = b \(\Rightarrow a^{2012}+b^{2012}=c^{2012}+d^{2012}\left(đpcm\right)\)

Với \(a+c=b+d\)

Mà \(a+b=c+d\)

\(\Rightarrow a+c+a+b=b+d+c+d\)

\(\Rightarrow2a=2d\Rightarrow a=d\Rightarrow a^{2012}=d^{2012}\left(1\right)\)

Lại có : \(a+c=b+d\)

\(\Rightarrow b=c\Rightarrow b^{2012}=c^{2012}\left(2\right)\)

Từ ( 1 ) ; ( 2 )

\(\Rightarrow a^{2012}+b^{2012}=c^{2012}+d^{2012}\left(đpcm\right)\)