CMO 1982

In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$ , 2, 3, and 4 ( $A_5 = A_1$ ). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$ .

$A_1$ $A_2$ $A_3$ $A_4$ $B_1$ $B_2$ $B_3$ $B_4$ $O$

Let $\mathbb{R}^n$ be the $n$ -dimensional Euclidean space. Determine the smallest number $g(n)$ of a points of a set in $\mathbb{R}^n$ such that every point in $\mathbb{R}^n$ is an irrational distance from at least one point in that set.

The altitudes of a tetrahedron $ABCD$ are extended externally to points $A'$ , $B'$ , $C'$ , and $D'$ , where $AA' = k/h_a$ , $BB' = k/h_b$ , $CC' = k/h_c$ , and $DD' = k/h_d$ . Here, $k$ is a constant and $h_a$ denotes the length of the altitude of $ABCD$ from vertex $A$ , etc. Prove that the centroid of tetrahedron $A'B'C'D'$ coincides with the centroid of $ABCD$ .