Given a circle with two chords $AB$, $CD$ that meet at $E$, let $M$ be a point of chord $AB$ other than $E$. Draw the circle through $D$, $E$, and $M$. The tangent line to the circle $DEM$ at $E$ meets the lines $BC$, $AC$ at $F$, $G$, respectively. Given $\frac{AM}{AB}=\lambda$, find $\frac{GE}{EF}$.
On a circle, $2n-1$ $(n\ge3)$ different points are given. Find the minimal natural number $N$ with the property that whenever $N$ of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly $n$ of the given $2n-1$ points.
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 $\mathbb Q^+$.
Two players $A$ and $B$ play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number $n_0\gt 1$ is given. Knowing $n_{2k}$, player $A$ may choose any $n_{2k+1}\in\mathbb N$ such that
$$n_{2k}\le n_{2k+1}\le n_{2k}^2.$$
Then player $B$ chooses a number $n_{2k+2}\in\mathbb N$ such that
$$\frac{n_{2k+1}}{n_{2k+2}}=p^r,$$
where $p$ is a prime number and $r\in\mathbb N$.
It is stipulated that player $A$ wins the game if he (she) succeeds in choosing the number 1990, and player $B$ wins if he (she) succeeds in choosing 1. For which natural numbers $n_0$ can player $A$ manage to win the game, for which $n_0$ can player $B$ manage to win, and for which $n_0$ can players $A$ and $B$ each force a tie?
Is there a 1990-gon with the following properties (i) and (ii)?
(i) All angles are equal;
(ii) The lengths of the 1990 sides are a permutation of the numbers $1^2,2^2,\ldots,1989^2,1990^2$.