Question 189
The numbers $2^3 - 2$, $3^3 - 3$, $4^3 - 4$, $\dots$, $(2n+1)^3 - (2n+1)$, where $n \ge 2$ is an integer, are written on a board. An operation consists in erasing three randomly chosen numbers $a$, $b$, $c$ from the board and replacing them with $\frac{abc}{ab+bc+ca}$. Several operations are performed until two numbers remain on the board. Show that the sum of the last two numbers left on the board is greater than $16$.
Gheorghe Rotariu
Gheorghe Rotariu