VN-TST 1999

Let an odd prime $p$ be a given number satisfying $2^h \neq 1 \pmod{p}$ for all $h < p-1, h \in \mathbb{N}^{*},$ and an even integer $a \in \left(\frac{p}{2},p \right).$ Let us consider the sequence $\{a_n\}^{\infty}_{n=0}$ defined by $a_0 = a$ and $a_{n+1} = p - b_n$ for $n = 0, 1, 2, \ldots$ , where $b_n$ is the greatest odd divisor of $a_n.$ Show that $\{a_n\}$ is periodical and find its least positive period.

Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that $f(x) = q \cdot g(x)$ for all $x \in R$ .

I. Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar.

II. How many polynomials of degree 1999 are there which have above mentioned property.

Let a convex polygon $H$ be given. Show that for every real number $a \in (0, 1)$ there exist 6 distinct points on the sides of $H$ , denoted by $A_1, A_2, \ldots, A_6$ clockwise, satisfying the conditions:

I. $(A_1A_2) = (A_5A_4) = a \cdot (A_6A_3)$ . II. Lines $A_1A_2, A_5A_4$ are equidistant from $A_6A_3$ .

(By $(AB)$ we denote vector $AB$ )

Let a sequence of positive reals $\{u_n\}^{\infty}_{n=1}$ be given. For every positive integer $n$ , let $k_n$ be the least positive integer satisfying:

$$\sum^{k_n}_{i=1} \frac{1}{i} \geq \sum^n_{i=1} u_i.$$

Show that the sequence $\left\{\frac{k_{n+1}}{k_n}\right\}$ has finite limit if and only if $\{u_n\}$ does.

Let a triangle $ABC$ inscribed in circle $\Gamma$ be given. Circle $\Theta$ lies in angle $Â$ of triangle and touches sides $AB, AC$ at $M_1, N_1$ and touches internally $\Gamma$ at $P_1$ . The points $M_2, N_2, P_2$ and $M_3, N_3, P_3$ are defined similarly to angles $B$ and $C$ respectively. Show that $M_1N_1, M_2N_2$ and $M_3N_3$ intersect each other at their midpoints.

Let a regular polygon with $p$ vertices be given, where $p$ is an odd prime number. At every vertex there is one monkey. An owner of monkeys takes $p$ peanuts, goes along the perimeter of polygon clockwise and delivers to the monkeys by the following rule: Gives the first peanut for the leader, skips the two next vertices and gives the second peanut to the monkey at the next vertex; skip four next vertices gives the second peanut for the monkey at the next vertex ... after giving the $k$ -th peanut, he skips the $2 \cdot k$ next vertices and gives $k+1$ -th for the monkey at the next vertex. He does so until all $p$ peanuts are delivered.

I. How many monkeys are there which does not receive peanuts? II. How many edges of polygon are there which satisfying condition: both two monkey at its vertex received peanut(s)?