\(\left(a+b+c\right)^2=0\Rightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=0\)
\(\Rightarrow ab+bc+ca=-\frac{2009}{2}\)
\(\Rightarrow\left(ab+bc+ca\right)^2=\frac{2009^2}{4}\)
\(\Rightarrow a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)=\frac{2009^2}{4}\)
\(\Rightarrow a^2b^2+b^2c^2+c^2a^2=\frac{2009^2}{4}\)
\(a^2+b^2+c^2=2009\)
\(\Rightarrow\left(a^2+b^2+c^2\right)^2=2009^2\)
\(\Rightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)=2009^2\)
\(\Rightarrow a^4+b^4+c^4=2009^2-2\left(a^2b^2+b^2c^2+c^2a^2\right)=2009^2-\frac{2.2009^2}{4}=\frac{2009^2}{2}\)
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