# Largest number obtained from products?

What is the largest number that can be obtained as the product of two or more positive integers that add up to 20?

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do they have to be distinct? – Jean-Sébastien Sep 30 '12 at 19:10
There are some IMO problems regarding things like this. – Beni Bogosel Sep 30 '12 at 19:24
The original problem doesn't state if they have to be distint, so I'm assuming it doesn't matter. – Ctrl Sep 30 '12 at 19:46

If a number $m\gt 4$ occurs in a decomposition (of $20$ in this case), the product can be increased by splitting $m$ suitably. It makes no difference whether $4$ is split as $2\cdot 2$ or not, so it might as well be. It is clear that it is no good to use any $1$'s.
Thus we can assume that all the numbers in our splitting of $20$ are $2$'s and/or $3$'s. Then note that $6=2+2+2=3+3$ but $3^2\gt 2^3$. So if there are three or more $2$'s in a splitting, we can do better.
Remark: If you want to prove that any number $m \gt 4$ should be split further, there are two cases to consider. (i) If $m=2k$, show that $k^2\gt 2k$ if $k\gt 2$. (ii) If $m=2k+1$, show that $k(k+1)\gt 2k+1$ if $k\ge 2$.