Ta có: \(a+b+c=0\)

\(\Leftrightarrow a+b=-c\)

\(\Leftrightarrow a^2+2ab+b^2=c^2\)

\(\Leftrightarrow a^2+b^2-c^2=-2ab\)

\(\Leftrightarrow a^4+b^4+c^4+2\left(a^2b^2-a^2c^2-b^2c^2\right)=4a^2b^2\)

\(\Leftrightarrow a^4+b^4+c^4=2\left(a^2b^2+a^2c^2+b^2c^2\right)\)

Ta lại có: \(a^2+b^2+c^2=2009\)

\(\Leftrightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)=2009^2\)

\(\Leftrightarrow2\left(a^4+b^4+c^4\right)=2009^2\)

\(\Leftrightarrow a^4+b^4+c^4=\frac{2009^2}{2}\)