We write 2017 as the sum of certain positive integers (for example, $2017=$ $1008+1009$ or $2017=458+776+793$). Then we multiply these integers. What is the largest product that we can obtain?
We observe that a decomposition into a sum of 2017 using a number $\geq 4$ can be beaten or matched by a decomposition by replacing this number with a sum of 2 and 3. Thus, the maximum value is achieved for a decomposition using only 2s and 3s (1s are useless here). Since $2^{3}<3^{2}$, we cannot have more than two 2s in an optimal decomposition, as we would improve it by replacing three 2s with two 3s. We find $4 \cdot 3^{671}$.