Problems to Give to Your Enemies

1. There are 9 cards numbered 1 to 9. Ada and Bob play a game where they alternate selecting cards (without replacement) until someone has three distinct cards summing to 15, with Ada going first. a) Prove that Bob does not have a winning strategy. b) Does Ada have a winning strategy? If so, what is it?
2. Are there positive integer solutions to $x^3 + y^3 = 9z^3$ other than $(1,2,1)$ and $(2, 1, 1)$?
3. Prove that for all non-negative real numbers $a, b, c$ we have $(a^2 + b^2 + c^2)^2 \ge 3(a^3b + b^3 c + c^3 a)$, and determine when equality holds.
4. Find all real numbers $r$ such that $n^r$ is an integer for all positive integers $n$.
5. Does there exist a positive integer $n$ such that $n, 2n, \ldots, 2015n$ all have the same multiset of nonzero digits?
6. Find all ordered triples of non-negative integers $(a,b,c)$ such that $a^2+2b+c$, $b^2+2c+a$, and $c^2+2a+b$ are all perfect squares.