VN-TST 1992

Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$ ) is not divided by $n$ , there exist two numbers $a_p, a_s$ ( $p \neq s$ ) such that $m + a_p - a_s$ is divided by $n$ .

Let a polynomial $f(x)$ be given with real coefficients and has degree greater or equal than 1. Show that for every real number $c > 0$ , there exists a positive integer $n_0$ satisfying the following condition: if polynomial $P(x)$ of degree greater or equal than $n_0$ with real coefficients and has leading coefficient equal to 1 then the number of integers $x$ for which $|f(P(x))| \leq c$ is not greater than degree of $P(x)$ .

Let $ABC$ a triangle be given with $BC = a$ , $CA = b$ , $AB = c$ ( $a \neq b \neq c \neq a$ ). In plane ( $ABC$ ) take the points $A'$ , $B'$ , $C'$ such that:

I. The pairs of points $A$ and $A'$ , $B$ and $B'$ , $C$ and $C'$ either all lie in one side either all lie in different sides under the lines $BC$ , $CA$ , $AB$ respectively;

II. Triangles $A'BC$ , $B'CA$ , $C'AB$ are similar isosceles triangles.

Find the value of angle $A'BC$ as function of $a, b, c$ such that lengths $AA', BB', CC'$ are not sides of an triangle. (The word "triangle" must be understood in its ordinary meaning: its vertices are not collinear.)

In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.

Find all pair of positive integers $(x, y)$ satisfying the equation

$$x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.$$

In a scientific conference, all participants can speak in total $2 \cdot n$ languages ( $n \geq 2$ ). Each participant can speak exactly two languages and each pair of two participants can have at most one common language. It is known that for every integer $k$ , $1 \leq k \leq n-1$ there are at most $k-1$ languages such that each of these languages is spoken by at most $k$ participants. Show that we can choose a group from $2 \cdot n$ participants which in total can speak $2 \cdot n$ languages and each language is spoken by exactly two participants from this group.