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 Principle of Inclusion-Exclusion,
, ,
By Principle of Inclusion-Exclusion,
, ,
By Principle of Inclusion-Exclusion,
, ,
By Principle of Inclusion-Exclusion,
See also
1991 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 29 |
Followed by Problem 30 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.