My 13 year old cousin has given me a question he solved at a math competition. I was trying to solve it myself but was unable to find the way.
The problem is this:
We have a list of these $9$ numbers - $1, 2, 3, 4, 5, 6, 7, 8, 9$. Remove two of them, sum them, and then add the sum back to the list. (for example by removing $7$ and $9$, the resulting list would be $1, 2, 3, 4, 5, 6, 8, 16)$. Now split the list into two smaller sublists, e.g $[1,2,3,4,5] = 120$ and $[6,8,16] = 768$. The product of the numbers of the two sublists, however, must be the same. What is the largest possible product in a sublist?
I was trying to go through integer factorization but I am really unable to even make the first step.
Consider $7$.
Suppose it is not removed. Then it must be in one of the sublists. The product of the numbers in that sublist must be a multiple of $7$. So the product of the numbers in the second sublist must be a multiple of $7$, too. However, the only number among $1,2,3,4,5,6,7,8,9$ that is a multiple of $7$ is $7$. Hence, the sum of the two removed numbers must be in the second sublist and it must be a multiple of $7$.
Similarly, either $5$ is not removed and the sum of the two removed numbers must be a multiple of $5$, or $5$ is removed.
Let us remove $2$ and $5$, which sum up to $7$. Adding $7$ back, we have $1,3,4,6,7,7,8,9$, which can be split into $1,3,4,6,7$ and $7,8,9$. The product of either list is $504$.
You can check $504$ is the wanted answer.