smc

Multiply out the LHS to get $100a^2+10ac+10ab+bc=100a(a+1)+bc$. Subtract $bc$ and factor to get $10a(10a+b+c)=10a(10a+10)$. Divide both sides by $10a$ and then subtract $10a$ to get $b+c=10$, giving an answer of $\boxed{b+c=10}$.