Question

Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A.$ Bob's list has $2024$ sets. Find the sum of the elements of $A.$

答案

Solution:

For a fixed $a \in A,$ the sets $B$ with maximum element $a$ consist of $a$ together with an arbitrary subset of $\{1, \ldots, a - 1\},$ so there are $2^{a-1}$ of them, and every set on Bob's list is counted exactly once by its maximum. Hence $\sum_{a \in A} 2^{a-1} = 2024.$

Since $2024 = 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 + 2^5 + 2^3$ and binary representations are unique, $A = \{11, 10, 9, 8, 7, 6, 4\}.$ The sum of the elements of $A$ is $11 + 10 + 9 + 8 + 7 + 6 + 4 = 55.$

2024 AIME II Problem 6

Solution:

Problem 6 in Other Years