A car travels from A to B covering half the distance at $$v_1$$ and the remaining half of the distance at $$v_2$$. This car then returns from B to A at $$v_3$$. If the time taken for the return journey from B to A is half of the time taken to go from A to B, find the relation between $$v_1$$, $$v_2$$ and $$v_3$$.

PLACEHOLDER ANSWER - the answer is a relation, not a number. Let $$AB = 2d$$. The forward time is $$\frac{d}{v_1} + \frac{d}{v_2}$$ and the return time is $$\frac{2d}{v_3}$$. Setting $$\frac{2d}{v_3} = \frac{1}{2}\left(\frac{d}{v_1} + \frac{d}{v_2}\right)$$ gives $$\frac{4}{v_3} = \frac{1}{v_1} + \frac{1}{v_2}$$, i.e. $$v_3 = \frac{4v_1 v_2}{v_1 + v_2}$$.