1990 IMO Problems

Problems of the 1990 IMO.

Day I

Problem 1

Chords $AB$ and $CD$ of a circle intersect at a point $E$ inside the circle. Let $M$ be an interior point of the segment $EB$ . The tangent line at $E$ to the circle through $D$ , $E$ , and $M$ intersects the lines $BC$ and $AC$ at $F$ and $G$ , respectively. If $\frac {AM}{AB} = t,$ find $\frac {EG}{EF}$ in terms of $t$ .

Solution

Problem 2

Let $n \geq 3$ and consider a set $E$ of $2n - 1$ distinct points on a circle. Suppose that exactly $k$ of these points are to be colored black. Such a coloring is good if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $n$ points from $E$ . Find the smallest value of $k$ so that every such coloring of $k$ points of $E$ is good.

Solution

Problem 3

Determine all integers $n > 1$ such that $\frac {2^n + 1}{n^2}$ is an integer.

Solution

Day II

Problem 4

Let ${\mathbb Q}^ +$ be the set of positive rational numbers. Construct a function $f : {\mathbb Q}^ + \rightarrow {\mathbb Q}^ +$ such that $f(xf(y)) = \frac {f(x)}{y}$ for all $x$ , $y$ in ${\mathbb Q}^ +$ .

Solution

Problem 5

Given an initial integer $n_0 > 1$ , two players, ${\mathcal A}$ and ${\mathcal B}$ , choose integers $n_1$ , $n_2$ , $n_3$ , $\ldots$ alternately according to the following rules :

I.) Knowing $n_{2k}$ , ${\mathcal A}$ chooses any integer $n_{2k + 1}$ such that \[ n_{2k} \leq n_{2k + 1} \leq n_{2k}^2. \] II.) Knowing $n_{2k + 1}$ , ${\mathcal B}$ chooses any integer $n_{2k + 2}$ such that \[ \frac {n_{2k + 1}}{n_{2k + 2}} \] is a prime raised to a positive integer power.

Player ${\mathcal A}$ wins the game by choosing the number 1990; player ${\mathcal B}$ wins by choosing the number 1. For which $n_0$ does :

a.) ${\mathcal A}$ have a winning strategy? b.) ${\mathcal B}$ have a winning strategy? c.) Neither player have a winning strategy?

Solution

Problem 6

Prove that there exists a convex 1990-gon with the following two properties :

a.) All angles are equal. b.) The lengths of the 1990 sides are the numbers $1^2$ , $2^2$ , $3^2$ , $\cdots$ , $1990^2$ in some order.

Solution

1990 IMO (Problems) • Resources
Preceded by 1989 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 1991 IMO
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1990 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6