# MAS114: Challenge Problem 3

### Problem

Suppose that we have some positive integers (not necessarily distinct) whose sum is $$100$$. How large can their product be? You should prove your answer is best.

### Solution

I got solutions from Alice Butler, Mohammad Tayyab Sajjad, Ben Andrews, James Mason and Oliver Feghali. Mostly they were slightly incomplete as proofs, but they had all of the ingredients: by combining bits and pieces I can produce a really nice one.

The best solution can't use a 1, as $$1+n$$ is always bigger than $$1\times n$$.

The best solution also can't use anything larger than 4, as $$2(n-2)$$ is bigger than $$2+(n-2)$$ for $$n\geq 5$$; similarly, it needn't use a 4, as $$2\times 2$$ is equal to $$2+2$$.

Therefore, we can only use numbers 2 and 3.

There can be at most two 2's, since $$3\times 3 > 2\times 2\times 2$$.

So we make our number with 3's, and either zero, one or two 2's. The only way of making a hundred is with thirty-two 3's and two 2's. This gives an answer of $$2^23^{32}$$.