VN-TST 1995

Let be given a triangle $ABC$ with $BC = a$ , $CA = b$ , $AB = c$ . Six distinct points $A_1$ , $A_2$ , $B_1$ , $B_2$ , $C_1$ , $C_2$ not coinciding with $A$ , $B$ , $C$ are chosen so that $A_1$ , $A_2$ lie on line $BC$ ; $B_1$ , $B_2$ lie on $CA$ and $C_1$ , $C_2$ lie on $AB$ . Let $\alpha$ , $\beta$ , $\gamma$ three real numbers satisfy $\overrightarrow{A_1A_2} = \frac {\alpha}{a}\overrightarrow{BC}$ , $\overrightarrow{B_1B_2} = \frac {\beta}{b}\overrightarrow{CA}$ , $\overrightarrow{C_1C_2} = \frac {\gamma}{c}\overrightarrow{AB}$ . Let $d_A$ , $d_B$ , $d_C$ be respectively the radical axes of the circumcircles of the pairs of triangles $AB_1C_1$ and $AB_2C_2$ ; $BC_1A_1$ and $BC_2A_2$ ; $CA_1B_1$ and $CA_2B_2$ . Prove that $d_A$ , $d_B$ and $d_C$ are concurrent if and only if $\alpha a + \beta b + \gamma c \neq 0$ .

Find all integers $k$ such that for infinitely many integers $n \ge 3$ the polynomial $$P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995$$ can be reduced into two polynomials with integer coefficients.

Find all integers $a$ , $b$ , $n$ greater than $1$ which satisfy $$\left(a^3 + b^3\right)^n = 4(ab)^{1995}$$

A graph has $n$ vertices and $\frac {1}{2}\left(n^2 - 3n + 4\right)$ edges. There is an edge such that, after removing it, the graph becomes unconnected. Find the greatest possible length $k$ of a circuit in such a graph.

For any nonnegative integer $n$ , let $f(n)$ be the greatest integer such that $2^{f(n)} | n + 1$ . A pair $(n, p)$ of nonnegative integers is called nice if $2^{f(n)} > p$ . Find all triples $(n, p, q)$ of nonnegative integers such that the pairs $(n, p)$ , $(p, q)$ and $(n + p + q, n)$ are all nice.

Consider the function $f(x) = \frac {2x^3 - 3}{3x^2 - 1}$ .

$1.$ Prove that there is a continuous function $g(x)$ on $\mathbb{R}$ satisfying $f(g(x)) = x$ and $g(x) > x$ for all real $x$ .

$2.$ Show that there exists a real number $a > 1$ such that the sequence $\{a_n\}$ , $n = 1, 2, \ldots$ , defined as follows $a_0 = a$ , $a_{n + 1} = f(a_n)$ , $\forall n\in\mathbb{N}$ is periodic with the smallest period $1995$ .