Find charge per unit length of conductors
This is a nice question...
Let's start with your last statement. It must be
$$\lambda_c=-(\lambda_a+\lambda_b)$$
because cylinder C has to cancel out the charges of cylinders A and B.
Now, think about the fields generated by the three charges. It looks like this:

Remember, that the charge on a conducting,  hollow body does not generate any E-field inside.
But the E-field of a charge inside a hollow body does not end at the surface, it expands to infinity.
This also is a result of Gauß' law, as it simply says that field times (closed) surface is charge inside that surface.
The superposition of all fields is the grey curve, and now you see that
$$V_b=\int_b^c\frac{(\lambda_a+\lambda_b)}{2\pi\varepsilon_0r}\,dr$$
and
$$V_a=\int_a^b\frac{\lambda_a}{2\pi\varepsilon_0r}\,dr+V_b$$
With given potential(difference)s, you can calculate \$\lambda_a\$ and \$\lambda_b\$.