I'm not sure what category these belong in. They're somewhat like puzzles.
3. A town has 2001 residents. Each resident has a non-negative amount of money (integer). If no two residents have the same amount of money and the combined sum of the balance of any 1000 residents is less than the combined sum of the balances of the remaining 1001 residents. Show that every resident has at least 1,000,000.
4. Show that for every integer n >= 2, the polynomial f(x) = x^n + (n^2+1)x + n has at least one root which is not a rational number