Invariants/monovariants: numbers on a board

Question

The numbers from $1$ through $2008$ are written on a blackboard. Every second, Dr. Math erases four numbers of the form $a, b, c, a+b+c$, and replaces them with the numbers $a+b, b+c, c+a$.

Prove that this can continue for at most $10$ minutes.

Answer 1

Observe that both the sum of the numbers and the sum of their squares is invariant under the process. Let $n=2008$.

Let $m$ be the number of integers on the board.

Then by Cauchy-Schwarz, we have $\dfrac{n(n+1)(2n+1)}{6} \geq \dfrac{n^2(n+1)^2}{4m}$, i.e. $m \geq \dfrac{3n(n+1)}{2(2n+1)}$ and hence the number of times the process can continue is at most $n-m\leq\dfrac{n^2-n}{2(2n+1)}$ which is less than 502 for $n=2008$.