1. Construct a 3 3 square. Prove that if 10 points are selected at random in the interior of the square, there must be at least one pair whose distance from each other is less than 2. 2. Prove that, for any positive numbers a and b, ab a+b \2. 3. Prove that, if x is an odd integer, then x2 1(mod 8) 4. Let m, n be two integers. Prove that m n(mod 15) if and only if m n(mod 3) and m n(mod 5). 5. Prove that every prime number larger than 3 can be written in the form p = 6k + 1 or p = 6k + 5 for some integer k. 6. Prove that (N Z) (Z N) = N N. 7. Let U be some universal set and let A U . Find a way of writing U U in terms of Cartesian products of A and Ac. Then prove your equation. 8. Let A and B be two sets. Find formulas for the power sets P(A B) and P(A B) in terms of P(A) and P(B). Then prove your equations. (Note, one formula is very easy, and the other may require some creativity!) 9. Find an interval, in closed form, that is equivalent to nZ+ [2 1n , 4 1n] and then prove that the two sets are equal. 10. Suppose you work at a jail with 100 jail cells, and the cells are numbered 1 through 100. One night, when all of the cells are locked, some mathematical minions enter the jail cell to cause mischief. They cause mathematical mischief in the following maniacal manner: The first minion switches the locks of the doors whose numbers are multiples of 1, and then The second minion switches the locks of the doors whose numbers are multiples of 2, and then The third minion switches the locks of the doors whose numbers are multiples of 3, and then . . . . . . The kth minion switches the locks of the doors whose numbers are multiples of k, and then . . . The 100th minion switches the locks of the doors whose numbers are multiples of 100. After all of this chaos, which cells are unlocked?

Using the proofs above to answer the questions down below

A proof that you found difficult, but eventually understood Your most "epic failure" on a proof (a proof that went disastrously wrong on your first attempt) A proof by contradiction A proof of set equality or inclusion A proof of an existential statement/proof by construction A proof by induction A proof on whether a function is injective and surjective