The Marathon

It seems as though Alice would win by a little more than $26$ seconds ahead of Bob, since she finishes each mile a second faster than him. However, we must also consider that Bob runs with varying speeds. In other words, he could run extremely fast for the first half of the mile, then much slower the second half, allowing Alice to catch up and get a second ahead. (Note: Bob finishes each MILE a second slower than Alice. He could easily be faster than Alice for $.2$ miles.) Using this logic, it is easy to see that if Bob runs fast enough, he can easily beat Alice's $26$ second head start the last $.2$ mile stretch of the marathon. Of course, if Bob runs slower for the first half, then Alice would easily win the marathon. Therefore, $\boxed{\text{either could win}}$ .

Because a marathon is 26.2 miles and Bob can run at any speed, just finishing 1 second slower than Alice at each time, this scenario is possible: Alice = 0.0000000001 mph Bob = 99999999 mph except he stops and waits for Alice at each mile.

Therefore, if there is 1+ possibility on each side, either could win.

We have no information about their relative speeds for the last 0.2 miles. Bob will start that last 0.2 miles 26 seconds behind Alice, and that's a lot of time to make up. But we have not information indicating it's not possible.