1991 AHSME Problems/Problem 30
Problem
For any set , let denote the number of elements in , and let be the number of subsets of , including the empty set and the set itself. If , , and are sets for which and , then what is the minimum possible value of ?
Solution 1
, so and are integral powers of and . Let , , and where Thus, the minimum value of is
Solution 2 (PIE)
As ,
As , ,
as and are integers, and
By the Principle of Inclusion-Exclusion,
, ,
By the Principle of Inclusion-Exclusion,
, ,
By the Principle of Inclusion-Exclusion,
, ,
By the Principle of Inclusion-Exclusion,
See also
1991 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 30 | |
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