International Mathematical Olympiad  – IMO 1997  Problems

Problem 1

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately 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) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$.

c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.

Problem 2

It is known that $ \angle BAC$ is the smallest angle in the 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$ at $ V$ and $ W$, respectively. The lines $ BV$ and $ CW$ meet at $ T$.

Show that $ AU = TB + TC$.

Alternative formulation:

Four different points $ A,B,C,D$ are chosen on a circle $ \Gamma$ such that the triangle $ BCD$ is not right-angled. Prove that:

(a) The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AD$ at certain points $ W$ and $ V,$ respectively, and that the lines $ CV$ and $ BW$ meet at a certain point $ T.$

(b) The length of one of the line segments $ AD, BT,$ and $ CT$ is the sum of the lengths of the other two.

Problem 3

Let $ x_1$, $ x_2$, $ \ldots$, $ x_n$ be real numbers satisfying the conditions:
\[ \left\{\begin{array}{cccc} |x_1 + x_2 + \cdots + x_n | & = & 1 & \ \\
|x_i| & \leq & \displaystyle \frac {n + 1}{2} & \ \textrm{ for }i = 1, 2, \ldots , n. \end{array} \right.
\]
Show that there exists a permutation $ y_1$, $ y_2$, $ \ldots$, $ y_n$ of $ x_1$, $ x_2$, $ \ldots$, $ x_n$ such that
\[ | y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.
\]

Problem 4

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

(a) there is no silver matrix for $ n = 1997$;

(b) silver matrices exist for infinitely many values of $ n$.

Problem 5

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} = b^a$.

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 \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.