Difference between revisions of "2022 AIME I Problems/Problem 12"
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− | . | + | .== Problem == |
+ | For any finite set <math>X</math>, let <math>| X |</math> denote the number of elements in <math>X</math>. Define | ||
+ | <cmath> | ||
+ | \[ | ||
+ | S_n = \sum | A \cap B | , | ||
+ | \] | ||
+ | </cmath> | ||
+ | where the sum is taken over all ordered pairs <math>(A, B)</math> such that <math>A</math> and <math>B</math> are subsets of <math>\left\{ 1 , 2 , 3, \cdots , n \right\}</math> with <math>|A| = |B|</math>. | ||
+ | For example, <math>S_2 = 4</math> because the sum is taken over the pairs of subsets | ||
+ | <cmath> | ||
+ | \[ | ||
+ | (A, B) \in \left\{ (\emptyset, \emptyset) , ( \{1\} , \{1\} ), ( \{1\} , \{2\} ) , ( \{2\} , \{1\} ) , ( \{2\} , \{2\} ) , ( \{1 , 2\} , \{1 , 2\} ) \right\} , | ||
+ | \] | ||
+ | </cmath> | ||
+ | giving <math>S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4</math>. | ||
+ | Let <math>\frac{S_{2022}}{S_{2021}} = \frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find the remainder when <math>p + q</math> is divided by | ||
+ | 1000. |
Revision as of 22:59, 17 February 2022
.== Problem == For any finite set , let denote the number of elements in . Define where the sum is taken over all ordered pairs such that and are subsets of with . For example, because the sum is taken over the pairs of subsets giving . Let , where and are relatively prime positive integers. Find the remainder when is divided by 1000.