Relativistic spaceships moving towards each other There are two spaceships, A and B, moving towards each other, such that they will eventually pass each other. At a point equidistant from both ships, their velocities are both 0.5c towards that point. 
Using relativistic velocity addition, the velocity of B in the reference frame of A is (1/(1.25))c, and vice versa. In the reference frame of A, a clock on B will be ticking slower than a clock on A, due to time dilation. Likewise, in the reference frame of B, a clock on A will be ticking slower than a clock on B, due to time dilation. 
At the point at which they pass each other, the pilots of the ships show their clocks to each other via windows on their respective ships. My question is, will each pilot read both clocks differently from the other pilot? Pilot A should observe that his clock has been moving faster than Pilot B's, while Pilot B should observe that his clock has been moving faster than Pilot A's. Is there a paradox here, or have I been careless about something? (Note that I have been careful to avoid any acceleration/deceleration in this problem.)
 A: I would not ordinarily answer a question like this, because these questions usually come from people who have made no effort.  It's clear from your comment on m4r35n357's answer that you are an exception to that rule, so I'm happy to provide the spacetime diagram.  You will find that it pays to get good at drawing these; as m4r35n357 says, they are always the best way to clear up confusion about relativity.

A: Draw a spacetime diagram with the two spaceships approaching the origin (your equidistant point) at 0.5c.  Read off from that diagram the (equal) elapsed proper times ($\tau^2 = t^2 - x^2$) for each ship when they meet (and can compare readings), and also the elapsed coordinate time (which will be longer).
Only now should you consider time dilation; it is an effect, not a cause!  You might begin to doubt the need for even considering it, as you have all the numbers you need from the spacetime diagram.
TL;DR, always start with a spacetime diagram.  Don't try to use Lorentz transforms, the velocity addition formula or time dilation/length contraction as your primary analysis.  This is the easy way to do SR!
