VN-TST 1997

Let $ABCD$ be a given tetrahedron, with $BC = a$ , $CA = b$ , $AB = c$ , $DA = a_1$ , $DB = b_1$ , $DC = c_1$ . Prove that there is a unique point $P$ satisfying $$PA^2 + a_1^2 + b^2 + c^2 = PB^2 + b_1^2 + c^2 + a^2 = PC^2 + c_1^2 + a^2 + b^2 = PD^2 + a_1^2 + b_1^2 + c_1^2$$ and for this point $P$ we have $PA^2 + PB^2 + PC^2 + PD^2 \ge 4R^2$ , where $R$ is the circumradius of the tetrahedron $ABCD$ . Find the necessary and sufficient condition so that this inequality is an equality.

There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:

1. from each town there are exactly $k$ direct routes to $k$ other towns;
2. if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.

Find the greatest real number $\alpha$ for which there exists a sequence of infinitive integers $(a_n)$ , ( $n = 1, 2, 3, \ldots$ ) satisfying the following conditions:

1. $a_n > 1997n$ for every $n \in\mathbb{N}^{*}$ ;
2. For every $n\ge 2$ , $U_n\ge a^{\alpha}_n$ , where $U_n = \gcd\{a_i + a_k | i + k = n\}$ .

The function $f : \mathbb{N} \to \mathbb{Z}$ is defined by $f(0) = 2$ , $f(1) = 503$ and $f(n + 2) = 503f(n + 1) - 1996f(n)$ for all $n \in\mathbb{N}$ . Let $s_1$ , $s_2$ , $\ldots$ , $s_k$ be arbitrary integers not smaller than $k$ , and let $p(s_i)$ be an arbitrary prime divisor of $f\left(2^{s_i}\right)$ , ( $i = 1, 2, \ldots, k$ ). Prove that, for any positive integer $t$ ( $t\le k$ ), we have $2^t \Big | \sum_{i = 1}^kp(s_i)$ if and only if $2^t | k$ .

Find all pairs of positive real numbers $(a, b)$ such that for every $n \in\mathbb{N}^*$ and every real root $x_n$ of the equation $4n^2x = \log_2(2n^2x + 1)$ we always have $a^{x_n} + b^{x_n} \ge 2 + 3x_n$ .

Let $n$ , $k$ , $p$ be positive integers with $2 \le k \le \frac {n}{p + 1}$ . Let $n$ distinct points on a circle be given. These points are colored blue and red so that exactly $k$ points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least $p$ red points. How many such colorings are there?