IMO 1996

We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $AB = 20, BC = 12$ . The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $\sqrt {r}$ . The task is to find a sequence of moves leading 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 is possible when $r = 73$ .

(c) Can the task be done when $r = 97$ ?

Let $P$ be a point inside a triangle $ABC$ such that $$\angle APB - \angle ACB = \angle APC - \angle ABC.$$ Let $D$ , $E$ be the incenters of triangles $APB$ , $APC$ , respectively. Show that the lines $AP$ , $BD$ , $CE$ meet at a point.

Let $p,q,n$ be three positive integers with $p + q < n$ . Let $(x_{0},x_{1},\cdots ,x_{n})$ be an $(n + 1)$ -tuple of integers satisfying the following conditions :

(a) $x_{0} = x_{n} = 0$ , and

(b) For each $i$ with $1\leq i\leq n$ , either $x_{i} - x_{i - 1} = p$ or $x_{i} - x_{i - 1} = - q$ .

Show that there exist indices $i < j$ with $(i,j)\neq (0,n)$ , such that $x_{i} = x_{j}$ .