\(S=\frac{a}{b+c}+1+\frac{b}{c+a}+1+\frac{c}{a+b}+1-3\)
\(S=\frac{a+b+c}{b+c}+\frac{a+b+c}{c+a}+\frac{a+b+c}{a+b}-3\)
\(S=\frac{2001}{b+c}+\frac{2001}{c+a}+\frac{2001}{a+b}-3\)
\(S=2001\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)-3\)
\(S=2001.\frac{1}{10}-3=\frac{1971}{10}\)