If $$a + \frac{1}{b}$$  = $$b + \frac{1}{c}$$  = $$c + \frac{1}{a}$$ $$\left( {a \ne b \ne c} \right)$$   then the value of abc is?

$$a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$$
To save your time assume values of a, b, c according to the question.
$$\eqalign{ & {\text{Let }}a = 2,{\text{ }}b = - 1\& c = \frac{1}{2} \cr & 2 + \frac{1}{{ - 1}} = - 1 + \frac{1}{{\frac{1}{2}}} = \frac{1}{2} + \frac{1}{2} \cr & \therefore abc = 2 \times - 1 \times \frac{1}{2} = - 1 \cr} $$