IMO 1990

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 $t$.

Problem 2. Let $n\ge 3$ 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.

Problem 3. Determine all integers $n>1$ such that $\frac{2^n+1}{n^2}$ is an integer.

Problem 4. 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 ${\mathbb{Q}}^{+}$.

Problem 5. Given an initial integer $n_0>1$, two players, $\mathcal{A}$ and $\mathcal{B}$, choose integers $n_1$, $n_2$, $n_3$, ... alternately according to the following rules:

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

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?

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