Math Problems?

[MICHAEL L]

Please SHOW YOUR WORK, NO GUESS AND CHECK!!!!

1. Given any three natural numbers, show that there are two of them, say a and b,
such that a^3b – b^3a is divisible by 30.
2. In triangle ABC, draw angle bisectors AD and CE, where D is on BC and E is
on AB. If angle B is 60 degrees, show that AC = CD + AE.
3. Let A, B, C and D be points of the plane. Let P be the disk for which AB is a
diameter, and Q the disk for which CD is a diameter. Assume that A, B do not
belong to Q and C, D do not belong to P. Prove that the segments AB and CD
are disjoint.
4. A tennis tournament has every participant playing every other participant. No
ties are allowed. A player A is superior if for every other player B, either A
beats B or there is a third player C such that A beats C and C beats B. Prove
that if there is only one superior player then he or she beats every other player.