CMR:Nếu a^4+b^4+c^4+d^4=4abcd và a,b,c,d>0 thì a=b=c=d

$\displaystyle  \begin{array}{{>{\displaystyle}l}}
a^{4} +b^{4} +c^{4} +d^{4} =4abcd\\
\Leftrightarrow a^{4} +b^{4} +c^{4} +d^{4} -4abcd=0\\
\Leftrightarrow a^{4} -2a^{2} b^{2} +b^{4} +c^{4} -2c^{2} d^{2} +d^{4} +2a^{2} b^{2} -4abcd+2c^{2} d^{2} =0\\
\Leftrightarrow \left( a^{2} -b^{2}\right)^{2} +\left( c^{2} -d^{2}\right)^{2} +2( ab-cd)^{2} =0\\
\left( a^{2} -b^{2}\right)^{2} \geqslant 0\forall a,b\\
\left( c^{2} -d^{2}\right)^{2} \geqslant 0\forall c,d\\
( ab-cd)^{2} \geqslant 0\forall a,b,c,d\\
DBXR\ \Leftrightarrow \begin{cases}
a^{2} -b^{2} =0 & \\
c^{2} -d^{2} =0 & \\
ab-cd=0 & 
\end{cases}\\
\Leftrightarrow \begin{cases}
a=\pm b & \\
c=\pm d & \\
ab=cd & 
\end{cases}\\
\Leftrightarrow a=b=c=d
\end{array}$