The symbols $ (a,b,\ldots,g)$ and $ [a,b,\ldots,g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $ a,b,\ldots,g$. For example, $ (3,6,18)=3$ and $ [6,15]=30$. Prove that \[ \frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}.\]

A given tetrahedron $ ABCD$ is isoceles, that is, $ AB=CD$, $ AC=BD$, $ AD=BC$. Show that the faces of the tetrahedron are acute-angled triangles.

A random selector can only select one of the nine integers $ 1,2,\ldots,9$, and it makes these selections with equal probability. Determine the probability that after $ n$ selections ($ n>1$), the product of the $ n$ numbers selected will be divisible by 10.

Let $ R$ denote a non-negative rational number. Determine a fixed set of integers $ a,b,c,d,e,f$, such that for every choice of $ R$, \[ \left| \frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right| < \left|R-\sqrt[3]{2}\right|.\]

A given convex pentagon $ ABCDE$ has the property that the area of each of five triangles $ ABC, BCD, CDE, DEA$, and $ EAB$ is unity (equal to 1). Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.

These problems are copyright $\copyright$ Mathematical Association of America.