International Mathematical Olympiad  – IMO 1996  Problems

Problem 1

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$?

Problem 2

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.

Problem 3

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that
\[ f(m + f(n)) = f(f(m)) + f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.
\]

Problem 4

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?

Problem 5

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
\[ R_{A} + R_{C} + R_{E}\geq \frac {P}{2}.
\]

Problem 6

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}$.