<p>1.) Yes, Real = x, fake = x - 1: 20x + 21x - 21 = 41x - 21. If he loses a real one, he will have 40x - 21 which are odd no matter what and wouldn’t balance. If he loses a fake one, he will have 40x - 20 which is even no matter what and would balance so yes it is possible.
2.) If they are moving at the same speed, then Old Junky would win because it only has to go 2/3 of the distance in the time New Sporty has to go the whole distance. Your wording is quite awkward so you may have left out part of it.
3.) 148, because if he took all of any two kinds, the total has to be 99 or below.
4.) If I understand it it is 0 because “The distance between any two of them is greater than d.”</p>