1996 IMO Problems

Problems of the 1996 IMO.

Day I

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 $r=97$ ?

Solution

Problem 2

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.

Solution

Problem 3

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

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

Solution

Day II

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?

Solution

Problem 5

Let $ABCDEF$ be a convex hexagon such that $AB$ is parallel to $DE$ , $BD$ 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

$R_{A}+R_{C}+R_{E} \ge \frac{P}{2}.$

Solution

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:

(i) $x_0=x_n=0$ ;

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

Show that there exists indices $i<j$ with $(i,j) \ne (0,n)$ , such that $x_i=x_j$ .

Solution

1996 IMO (Problems) • Resources
Preceded by 1995 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1997 IMO
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1996 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

See Also