PL-OM 1994

Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$ , $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.

Let be given two parallel lines $k$ and $l$ , and a circle not intersecting $k$ . Consider a variable point $A$ on the line $k$ . The two tangents from this point $A$ to the circle intersect the line $l$ at $B$ and $C$ . Let $m$ be the line through the point $A$ and the midpoint of the segment $BC$ . Prove that all the lines $m$ (as $A$ varies) have a common point.

$k$ is a fixed positive integer. Let $a_n$ be the number of maps $f$ from the subsets of $\{1, 2, ... , n\}$ to $\{1, 2, ... , k\}$ such that for all subsets $A, B$ of $\{1, 2, ... , n\}$ we have $f(A \cap B) = \min (f(A), f(B))$ . Find $\lim_{n \to \infty} \sqrt[n]{a_n}$ .

$m, n$ are relatively prime. We have three jugs which contain $m$ , $n$ and $m+n$ liters. Initially the largest jug is full of water. Show that for any $k$ in $\{1, 2, ... , m+n\}$ we can get exactly $k$ liters into one of the jugs.

A parallelopiped has vertices $A_1, A_2, ... , A_8$ and center $O$ . Show that: $$4 \sum_{i=1}^8 OA_i ^2 \leq \left(\sum_{i=1}^8 OA_i \right) ^2$$

The distinct reals $x_1, x_2, ... , x_n$ ,( $n > 3$ ) satisfy $\sum_{i=1}^n x_i = 0$ , $\sum_{i=1}^n x_i ^2 = 1$ . Show that four of the numbers $a, b, c, d$ must satisfy: $$a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd$$.