Problem:
How many nonempty subsets of have the property that the number of elements in is equal to the least element of ? For example, satisfies the condition.
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Any nonempty subset satisfying this property with minimum value must contain elements greater than . This implies that , and there are such subsets. The total number of subsets with the required property is therefore
Note: This problem illustrates an identity relating the sum of the elements on a "shallow" diagonal in Pascal's triangle to the Fibonacci numbers.
The problems on this page are the property of the MAA's American Mathematics Competitions