\(R=2^{2010}-\left[2^{2009}+2^{2008}+...+2^1+2^0\right]\)
\(2^{2010}-R=2^0+2^1+...+2^{2008}+2^{2009}\)
\(2^{2011}-2R=2^1+2^2+...+2^{2009}+2^{2010}\)
\(\left(2^{2011}-2R\right)-\left(2^{2010}-R\right)=2^{2010}-1\)
\(2^{2010}-R=2^{2010}-1\)
\(R=2^{2010}-2^{2010}+1=0+1=1\)