2018 AIME I Problems/Problem 9: Difference between revisions
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==Solutions== | ==Solutions== | ||
==Solution | ==Solution 1== | ||
This problem is tricky because it is the capital of a few "bashy" calculations. Nevertheless, the process is straightforward. Call the set <math>\{a, b, c, d\}</math>. | This problem is tricky because it is the capital of a few "bashy" calculations. Nevertheless, the process is straightforward. Call the set <math>\{a, b, c, d\}</math>. | ||
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We can look for solutions by listing possible <math>a</math> values and filling in the blanks. Start with <math>a=4</math>, as that is the minimum. We find <math>\{4, 12, 20, ?\}</math>, and likewise up to <math>a=15</math>. But we can't have <math>a=8</math> or <math>a=12</math> because <math>a=b</math> or <math>a=c</math>, respectively! Now, it would seem like there are <math>10</math> values for <math>a</math> and <math>17</math> unique values for each <math>?</math>, giving a total of <math>170</math>, 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 <math>a=8</math> and 3 pairs about <math>a=12</math>, meaning we lose <math>6</math>. That's <math>164</math> for Case 2. | We can look for solutions by listing possible <math>a</math> values and filling in the blanks. Start with <math>a=4</math>, as that is the minimum. We find <math>\{4, 12, 20, ?\}</math>, and likewise up to <math>a=15</math>. But we can't have <math>a=8</math> or <math>a=12</math> because <math>a=b</math> or <math>a=c</math>, respectively! Now, it would seem like there are <math>10</math> values for <math>a</math> and <math>17</math> unique values for each <math>?</math>, giving a total of <math>170</math>, 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 <math>a=8</math> and 3 pairs about <math>a=12</math>, meaning we lose <math>6</math>. That's <math>164</math> for Case 2. | ||
Total gives <math>\boxed{210}</math> | Total gives <math>\boxed{210}</math>. | ||
-expiLnCalc | -expiLnCalc | ||
Revision as of 19:22, 7 March 2018
Find the number of four-element subsets of 
with the property that two distinct elements of a subset have a sum of 
, and two distinct elements of a subset have a sum of 
. For example, 
and 
are two such subsets.
Solutions
Solution 1
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
![\[\{1, 15, 9, 15\}, \{2, 14, 10, 14\}, \{3, 13, 11, 13\}, \{4, 16, 4, 20\}, \{5, 11, 5, 19\},\]](http://latex.artofproblemsolving.com/4/a/9/4a935e4e076d965e072e64cc0dac959487372318.png)
![\[\{5, 11, 11, 13\}, \{6, 10, 6, 18\}, \{6, 10, 10, 14\}, \{7, 9, 9, 15\}, \{7, 9, 7, 17\}\]](http://latex.artofproblemsolving.com/6/1/f/61f420cd5557117b1cf09fabdd4b4c4952fb3ab9.png)
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 
.
-expiLnCalc
