VN-TST 2004

Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$ , satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \forall i, j$ from $\{1, 2,\ldots , 2004\}$ , where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$ . Find the least positive integer $k$ for which in every $k$ -subset of $S$ , having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.

Find all real values of $\alpha$ , for which there exists one and only one function $f: \mathbb{R} \mapsto \mathbb{R}$ and satisfying the equation $$f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y$$ for all $x, y \in \mathbb{R}$ .

In the plane, there are two circles $\Gamma_1, \Gamma_2$ intersecting each other at two points $A$ and $B$ . Tangents of $\Gamma_1$ at $A$ and $B$ meet each other at $K$ . Let us consider an arbitrary point $M$ (which is different of $A$ and $B$ ) on $\Gamma_1$ . The line $MA$ meets $\Gamma_2$ again at $P$ . The line $MK$ meets $\Gamma_1$ again at $C$ . The line $CA$ meets $\Gamma_2$ again at $Q$ . Show that the midpoint of $PQ$ lies on the line $MC$ and the line $PQ$ passes through a fixed point when $M$ moves on $\Gamma_1$ .

[Moderator edit: This problem was also discussed on http://www.mathlinks.ro/Forum/viewtopic.php?t=21414 .]

Let us consider a convex hexagon ABCDEF. Let $A_1, B_1,C_1, D_1, E_1, F_1$ be midpoints of the sides $AB, BC, CD, DE, EF,FA$ respectively. Denote by $p$ and $p_1$ , respectively, the perimeter of the hexagon $A B C D E F$ and hexagon $A_1B_1C_1D_1E_1F_1$ . Suppose that all inner angles of hexagon $A_1B_1C_1D_1E_1F_1$ are equal. Prove that $$p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .$$ When does equality hold ?

Let $S$ be the set of positive integers in which the greatest and smallest elements are relatively prime. For natural $n$ , let $S_n$ denote the set of natural numbers which can be represented as sum of at most $n$ elements (not necessarily different) from $S$ . Let $a$ be greatest element from $S$ . Prove that there are positive integer $k$ and integers $b$ such that $|S_n| = a \cdot n + b$ for all $n > k$ .