VN-TST 2001

Let a sequence of integers $\{a_n\}$ , $n \in \mathbb{N}$ be given, defined by

$$a_0 = 1, a_n= a_{n-1} + a_{[n/3]}$$

for all $n \in \mathbb{N}^{*}$ .

Show that for all primes $p \leq 13$ , there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$ . (Here $[x]$ denotes the integral part of real number $x$ ).

In the plane let two circles be given which intersect at two points $A, B$ ; Let $PT$ be one of the two common tangent line of these circles ( $P, T$ are points of tangency). Tangents at $P$ and $T$ of the circumcircle of triangle $APT$ meet each other at $S$ . Let $H$ be a point symmetric to $B$ under $PT$ . Show that $A, S, H$ are collinear.

Some club has 42 members. It's known that among 31 arbitrary club members, we can find one pair of a boy and a girl that they know each other. Show that from club members we can choose 12 pairs of knowing each other boys and girls.

Let's consider the real numbers $a, b, c$ satisfying the condition

$$21 \cdot a \cdot b + 2 \cdot b \cdot c + 8 \cdot c \cdot a \leq 12.$$

Find the minimal value of the expression

$$P(a, b, c) = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$

Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$ the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$ . We paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then points $B(x_1, y_1, z_1)$ for which $x_1 \leq x_0, y_1 \leq y_0$ and $z_1 \leq z_0$ could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.

Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition

$$0 < a_{n+1} - a_n \leq 2001$$

for all $n \in \mathbb{N}^{*}$

Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.