IMO 1990

Official IMO 1990 problems  ·  1/6 solved.

Problem 1   solved · 1m49s · Solution →

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 $n \geq 3$ 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.

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 $f : {\mathbb Q}^ + \rightarrow {\mathbb Q}^ +$ 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 $n_1$, $n_2$, $n_3$, $\ldots$ 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 $$ \dfrac{n_{2k + 1}}{n_{2k + 2}} $$ 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 $1^2$, $2^2$, $3^2$, $\cdots$, $1990^2$ in some order.