1997 IMO Problems

Problems of the 1997 IMO.

Day I

Problem 1

In the plane the points with integer coordinates are the vertices of unit squares. The squares are colored alternatively black and white (as on a chessboard).

For any pair of positive integers $m$ and $n$ , consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$ , lie along edges of the squares.

Let $S_{1}$ be the total area of the black part of the triangle and $S_{2}$ be the total area of the white part.

Let $f(m,n)=|S_{1}-S_{2}|$

(a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd.

(b) Prove that $f(m,n) \le \frac{1}{2} max\left\{ m,n \right\}$ for all $m$ and $n$ .

(c) Show that there is no constant $C$ such that $f(m,n)<C$ for all $m$ and $n$ .

Solution

Problem 2

The angle at $A$ is the smallest angle of triangle $ABC$ . The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$ . The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ and $V$ and $W$ , respectively. The lines $BV$ and $CW$ meet at $T$ . Show that.

Solution

Problem 3

Let $x_{1}$ , $x_{2}$ ,..., $x_{n}$ be real numbers satisfying the conditions

$|x_{1}+x_{2}+...+x_{n}|=1$

and

$|x_{i}| \le \frac{n+1}{2}$ , for $i=1,2,...,n$

Show that there exists a permutation $y_{1}$ , $y_{2}$ ,..., $y_{n}$ of $x_{1}$ , $x_{2}$ ,..., $x_{n}$ such that

$|y_{1}+2y_{2}+...+ny_{n}|\le \frac{n+1}{2}$

Solution

Day II

Problem 4

An $n \times n$ matrix whose entries come from the set $S={1,2,...,2n-1}$ is called a $\textit{silver}$ matrix if, for each $i=1,2,...,n$ , the $i$ th row and the $i$ th column together contain all elements of $S$ . Show that

(a) there is no $\textit{silver}$ matrix for $n=1997$ ;

(b) $\textit{silver}$ matrices exist for infinitely many values of $n$ .

Solution

Problem 5

Find all pairs $(a,b)$ of integers $a,b \ge 1$ that satisfy the equation

$a^{b^{2}}=b^{a}$

Solution

Problem 6

For each positive integer $n$ , let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$ , because the number 4 can be represented in the following four ways:

$4;2+2;2+1+1;1+1+1+1$

Prove that, for any integer $n \ge 3$ ,

$2^{n^{2}/4}<f(2^{n})<2^{n^{2}/2}$ .

Solution

1997 IMO (Problems) • Resources
Preceded by 1996 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1998 IMO
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1997 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

See Also