2018 AIME I Problems/Problem 9
How many distinct subsets of 
are there such that there are two distinct elements that sum to 
and two distinct elements that sum to 
? Two examples are 
and 
.
Feel free to correct this problem if the wording isn't verbatim (I know it isn't).
Solutions
Solution Exclusion Awareness
This problem is tricky because it is the capital of a few "bashy" calculations. Nevertheless, the process is straightforward. Call the set 
.
Note that there are only two cases: 1 where 
and 
or 2 where and 
. Also note that there is no overlap between the two situations! This is because if they overlapped, adding the two equations of both cases and canceling out gives you 
, which is absurd.
Case 1.
This is probably the simplest: just make a list of possible combinations for 
and 
. We get 
for the first and 
for the second. That appears to give us 
solutions, right? NO. Because elements can't repeat, take out the supposed sets of 
. That's ten cases gone. So 
for Case 1.
Case 2.
We can look for solutions by listing possible 
values and filling in the blanks. Start with 
, as that is the minimum. We find 
, and likewise up to 
. But we can't have 
or 
because 
or 
, respectively! Now, it would seem like there are 
values for and 
unique values for each 
, giving a total of 
, but that is once again not true because there are some repeated values! We can subtract 1 from all pairs of sets that have two elements in common, because those can give us identical sets. There are 3 pairs about and 3 pairs about , meaning we lose 
. That's 
for Case 2.
Total gives 
. Lesson for this problem: Never be scared to attempt an AIME problem. You will oftentimes get it in ~10 minutes.
