International Mathematical Olympiad 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 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 $AP, BD, CE$ meet at a point.

Let $S$ denote the set of nonnegative integers. Find all functions $f$ from $S$ to itself such that

$$f(m + f(n)) = f(f(m)) + f(n) \qquad \forall m, n \in S.$$

The positive integers $a$ and $b$ are such that the numbers $15a + 16b$ and $16a - 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$, $BC$ is parallel to $EF$, and $CD$ is parallel to $FA$. 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

$$R_A + R_C + R_E \geq \frac{P}{2}.$$

Let $p, q, n$ be three positive integers with $p + q < n$. Let $(x_0, x_1, \ldots, x_n)$ be an $(n + 1)$-tuple of integers satisfying the following conditions:

(a) $x_0 = x_n = 0$.

(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 $x_i = x_j$.