Given the following NP-complete problem:
PARTITION
Input: A list of positive integers $a_1, a_2, \dots, a_n$.
Question: Can the list be partitioned into $2$ parts, $A_1$ and $A_2$, such that the sum of each part is the same?
Now I want to reduce PARTITION to the following problem:
TRIPLE-SUM
Input: A list of positive integers $a_1, a_2, \dots, a_n$.
Question: Can the list be partitioned into $2$ parts, $A_1$ and $A_2$, such that the sum of the numbers in $A_1$ is exactly three times the sum of the numbers in $A_2$?
I'm struggling to see how the transformation can take place. I've tried transforming the list $a_1, a_2, \dots, a_n$ by multiplying each element by $3$ (stupid... I know...) but everything I tried failed. Is there a way to perform this reduction? Does it even require to modify the input list?