Day 1
                               July 10, 1996
			  Time Alloted -- 4.5 hours

1. Let ABCD be a rectangular board, with AB = 20 and BC = 12.  The board is
   divided into 20x12 squares.  Let r be a given positive integer.  A coin can
   be moved from one square to another iff the distance between the centres of
   the two squares is sqrt(r).  The task is to find a sequence of moves taking
   the coin from the square with A as a vertex to the square with B as a
   vertex.

   a) Show that the task cannot be done if r is divisible by 2 or 3.
   b) Prove that the task can be done if r = 73.
   c) Can the task be done when r = 97?

2. Let P be a point inside triangle ABC such that angle(APB) - angle(ACB) =
   angle(APC) - angle(ABC).  Let D and E be the incentres of triangles APB and
   APC respectively.  Show that AP, BD, and CE meet at a point.

3. Let S = {0, 1, 2, ...}.  Find all functions f defined on S taking their
   values in S such that f(m + f(n)) = f(f(m)) + f(n) for all m and n in S.

                                   Day 2
                               July 11, 1996
			  Time Alloted -- 4.5 hours

4. The positive integers a and b are such that the numbers 15a + 16b and
   16a - 15b are both squares of positive integers.  Find the least possible
   value that can be taken by the minimum of these two squares.

5. Let ABCDEF be a convex hexagon, such that AB||ED, BC||FE, and CD||AF.  Let
   R_A, R_C, and R_E denote the circumradii of triangles FAB, BCD, and DEF
   respectively, and let p denote the perimeter of the hexagon.
   Prove that R_A + R_C + R_E >= p/2.

6. Let n, p, and q be positive integers with n > p + q.  Let x_0, x_1, ...,
   x_n be integers satisfying the following conditions:

   a) x_0 = x_n = 0, and
   b) For each integer i, 1 <= i <= n, either x_i - x_{i-1} = p or
      x_i - x_{i-1} = -q.  
   
   Show that there exists a pair (i,j) of indices with
   i < j, and (i,j) != (0,0), such that x_i = x_j.