Question 72
Let $m \ge 2$ be a positive integer. Consider the $2m$ numbers
$$
1 \cdot 2, 2 \cdot 3, 3 \cdot 4, \dots, 2m(2m + 1).
$$
A move consists of choosing three numbers $a, b, c$, and replacing them with the single number
$$
\frac{abc}{ab + bc + ca}.
$$
After $m-1$ such moves, only two numbers will remain. Supposing one of these is $\frac{4}{3}$, show that the other exceeds 4.
$$
1 \cdot 2, 2 \cdot 3, 3 \cdot 4, \dots, 2m(2m + 1).
$$
A move consists of choosing three numbers $a, b, c$, and replacing them with the single number
$$
\frac{abc}{ab + bc + ca}.
$$
After $m-1$ such moves, only two numbers will remain. Supposing one of these is $\frac{4}{3}$, show that the other exceeds 4.