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.