a+b+c=4034=p
\(\frac{a}{c+b}+\frac{b}{a+c}+\frac{c}{a+b}=\frac{p-\left(c+b\right)}{c+b}+\frac{p-\left(a+c\right)}{a+c}+\frac{p-\left(a+b\right)}{a+b}\)
\(=\frac{p}{c+b}+\frac{p}{a+c}+\frac{p}{a+b}-3=p\left[\frac{1}{c+b}+\frac{1}{a+c}+\frac{1}{a+b}\right]-3\)
\(=4034.\frac{1}{2}-3=2017-3=2014\)