All this talk of infinite slices made we wonder if I had it wrong but I'm pretty sure you don't want that as the solution. As you approach infinite slices you'll eat the same percentage of the pie as your friend's per slice time is a percentage of the combined value of both your per slice times. Take my x = 10, y = 35 example. The y value is 77.7777...% of the combined time (35/45).
In the 5 slice solution you get 80% of the pie. Now let's divide up into more slices:
If the total slices is 6, then the time to eat them (t) all will be:
t/x + t/y = 6
Solving for t gives us t = 6 / (1/x + 1/y) which with our x and y values gives us 46.666... t/x is 4.666 and t/y is 1.333. Since your friend would start his second piece before you could start your 5th (you know this because your partial piece time of .666 isn't higher than the .777 required to get there first) he gets 2 and you get 4 so your percentage of the pie is worse (4/6 or 66.666%) than in the 5 piece scenario.
Going up to 7 slices you get 5 of the 7 for 71.43%. At 8 you get 72.73%, etc... The pattern you get is the percentage you get oscillating around 77.777% with the variations approaching 0 as the number of slices increases. So infinite slices isn't the answer for maximum percentage of the pie, you have to take advantage of the fact that as long as you get to the last slice before your friend you can take as long as you need to finish it.
Damn. Way too much thought on a math problem.
