2. (agora scholarships) What is the maximum number of regions the plane can be split into by n lines. (Same problem for n circles, n planes, n spheres) Example: For n = 4 we can get 7 regions.
3. What is the maximum product you can obtain from a set of positive natural numbers that sum up to n.
4. (romanian national olympiad, 10th grade 2000) How many ways can you tile a 3xn rectangle with dominoes.

5.

5. (romanian IOI selection 1999)

==code(c) |

  for i1 = 1 to n
    for i2 = i1 to n
      for i3 = i2 to n

  for i1 = 1,n
    for i2 = i1,n
      for i3 = i2,n

        …

         for ik = ik - 1 to  n

         for ik = ik - 1,n

             print ‘*’
==
How many stars will be printed for a given n and k.

6. How many ways can k rooks on a nxn chessboard so they don’t attack eachother.
7. How many permutations of length n have no fixed points. A fixed point for a permutation p is p(i) = i

6. How many ways can k rooks be placed on a nxn chessboard so that they don’t attack each other.
7. (romanian county olympiad, 11th grade 2000) How many permutations of length n have no fixed points. A fixed point for a permutation p is p(i) = i

8. ()(())
9. How many ways can you go from 0,0 to n, m.
non palindromes