International Mathematical Olympiad 1990

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 $t$.

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.

Determine all integers $n > 1$ such that

$$\frac{2^n + 1}{n^2}$$

is an integer.

Let $\mathbb{Q}^+$ be the set of positive rational numbers. Construct a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that

$$f(xf(y)) = \frac{f(x)}{y}$$

for all $x, y$ in $\mathbb{Q}^+$.

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:

Knowing $n_{2k}$, $\mathcal{A}$ chooses any integer $n_{2k+1}$ such that

$$n_{2k} \leq n_{2k+1} \leq n_{2k}^2.$$

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?

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 $1^2, 2^2, 3^2, \ldots, 1990^2$ in some order.