Sounds impossible, doesn't it? If Rarity is going to fall behind by a second every mile, how can she ever catch up? But of course, there MUST be a way, or else we wouldn't have this puzzle.
First, think about how Rarity can run at different speeds, yet cover every mile in the same amount of time. She does this by sprinting for the first part of a mile, then jogging for the rest of it. Repeat for every mile: basically, she runs x of every mile, then jogs for the remainder.
Let's adapt the strategy to suit Rarity's needs. The race is 26.2 miles long. So if she loses a second every mile, after 26 miles she'll be 26 seconds behind, so we need to make up 26 seconds in 0.2 miles.
Make the strategy so that Rarity should run 0.2 miles at top speed, then jog for the rest of the mile.
Let x + y = 8 min, 1 sec, or 481 seconds. The important thing here is to focus on time, not distance. If Steady Pace runs at the constant speed of a mile every eight minutes, he's expected to finish a 26.2 mile marathon in 12,576 seconds (when you convert the minutes). So to beat him, even by a hair, we need to run this race in 12,575 seconds.
Rarity will be running the 0.2 mile cycle 27 times, and the remainder only 26 times, so plugged into a formula, we get:
27x +26y = 12,575. Our previous formula shows that y = 481 - x, so we sub y with that, and happily, we can cancel it down, resulting in x = 69 seconds, and y = 412 seconds.
So if Rarity sprints 0.2 miles in 69 seconds, and the remaining mile in 412 seconds, she'll lose every mile versus mile comparison, but the final 0.2 mile sprint should allow Rarity to just catch Steady Pace and win!