USAMO 1972

Let $(\mathrm{a}, \mathrm{b}, \ldots, \mathrm{k})$ denote the greatest common divisor of the integers $\mathrm{a}, \mathrm{b}, \ldots \mathrm{k}$ and $[\mathrm{a}, \mathrm{b}, \ldots, \mathrm{k}]$ denote their least common multiple. Show that for any positive integers $\mathrm{a}, \mathrm{b}, \mathrm{c}$ we have $(\mathrm{a}, \mathrm{b}, \mathrm{c})^{2}[\mathrm{a}, \mathrm{b}][\mathrm{b}, \mathrm{c}][\mathrm{c}, \mathrm{a}]=[\mathrm{a}, \mathrm{b}$, $c]^{2}(a, b)(b, c)(c, a)$.

A tetrahedron has opposite sides equal. Show that all faces are acute-angled.

$\mathrm{n}$ digits, none of them 0 , are randomly (and independently) generated, find the probability that their product is divisible by 10 .

Let $k$ be the real cube root of 2. Find integers A, B, C, a, b, c such that $\left|\left(A x^{2}+B x+C\right) /\left(a x^{2}+b x+c\right)-k\right|<\mid$ $\mathrm{x}-\mathrm{k} \mid$ for all non-negative rational $\mathrm{x}$.

A pentagon is such that each triangle formed by three adjacent vertices has area 1. Find its area, but show that there are infinitely many incongruent pentagons with this property.