VN-TST 1993

We call a rectangle of size $2 \times 3$ (or $3 \times 2$ ) without one cell in corner a $P$ -rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$ ) without two cells in opposite (under center of rectangle) corners a $S$ -rectangle. Using some squares of size $2 \times 2$ , some $P$ -rectangles and some $S$ -rectangles, one form one rectangle of size $1993 \times 2000$ (figures don't overlap each other). Let $s$ denote the sum of numbers of squares and $S$ -rectangles used in such tiling. Find the maximal value of $s$ .

A sequence $\{a_n\}$ is defined by: $a_1 = 1, a_{n+1} = a_n + \dfrac{1}{\sqrt{a_n}}$ for $n = 1, 2, 3, \ldots$ . Find all real numbers $q$ such that the sequence $\{u_n\}$ defined by $u_n = a_n^q$ , $n = 1, 2, 3, \ldots$ has nonzero finite limit when $n$ goes to infinity.

THERE MIGHT BE A TYPO!

Let's consider the real numbers $x_1, x_2, x_3, x_4$ satisfying the condition

$$\dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1$$

Find the maximal and the minimal values of expression:

$$A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2$$

Let $H$ , $I$ , $O$ be the orthocenter, incenter and circumcenter of a triangle. Show that $2 \cdot IO \geq IH$ . When does the equality hold ?

Let an integer $k > 1$ be given. For each integer $n > 1$ , we put

$$f(n) = k \cdot n \cdot \left(1-\frac{1}{p_1}\right) \cdot \left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_r}\right)$$

where $p_1, p_2, \ldots, p_r$ are all distinct prime divisors of $n$ . Find all values $k$ for which the sequence $\{x_m\}$ defined by $x_0 = a$ and $x_{m+1} = f(x_m), m = 0, 1, 2, 3, \ldots$ is bounded for all integers $a > 1$ .

Let $n$ points $A_1, A_2, \ldots, A_n$ , ( $n>2$ ), be considered in the space, where no four points are coplanar. Each pair of points $A_i, A_j$ are connected by an edge. Find the maximal value of $n$ for which we can paint all edges by two colors - blue and red such that the following three conditions hold:

I. Each edge is painted by exactly one color. II. For $i = 1, 2, \ldots, n$ , the number of blue edges with one end $A_i$ does not exceed 4. III. For every red edge $A_iA_j$ , we can find at least one point $A_k$ ( $k \neq i, j$ ) such that the edges $A_iA_k$ and $A_jA_k$ are blue.