Here are the IMO'96 problems as posted by Carl Miller:


First day (4 1/2 hours; 7 points per problem)

1. Let ABCD be a rectangular board with |AB| = 20, |BC| = 12.  The board
is divided into 20 x 12 unit squares.  Let r be a given positive integer.
A coin can be moved from one square to another if and only if the distance
between the centres of the two squares is [root] r.  The task is to find a
sequence of moves taking the coin from the square which has A as a vertex
to the square which has 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

          <APB - <ACB =3D <APC - <ABC

Let D, E be the incentres of triangles APB, APC respectively.  Show that
AP, BD, and CE meet at a point.


3. Let S =3D {0, 1, 2, 3, ...} be the set of non-negative integers.  Find
all functions defined on S and taking their values in S such that

          f(m + f(n)) = f(f(m)) + f(n) for all m, n, in S.


Second day (4 1/2 hours; 7 points per problem)

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 is parallel to ED, BC is
parallel to FE and CD is parallel to AF.  Let R_A, R_C, R_E denote the
circumradii of triangles FAB, BCD, 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, 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;

   (b) for each integer i with 1 <= i <= n,

          either x_i - x_(i-1) = p or x_i - x_(i-1) = -q.

Show that there eixsts a pair (i,j) of indices with i < j and (i,j) [not
equal to] (0,n) such that x_i = x_j.


The exam was quite hard this year... only one person got a 42 (from
Romania).  Romania won with 187, the US was 2nd with 185, and the next
highest score was 167.