Question

Let set $\mathcal{A}$ be a $90$ -element subset of $\{1, 2, 3, \ldots, 100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of possible values of $S.$

答案

Solution:

The smallest possible sum is $1 + 2 + \cdots + 90 = 4095,$ and the largest is $11 + 12 + \cdots + 100 = 4995.$

Every integer in between also occurs. Suppose $\mathcal{A}$ has sum $S \le 4994,$ and let $k$ be the smallest element of $\mathcal{A}$ with $k + 1 \notin \mathcal{A}.$ If $k$ were $100,$ then $\mathcal{A}$ would be a block of consecutive integers ending at $100,$ namely $\{11, \ldots, 100\},$ whose sum exceeds $4994.$ So $k \ne 100,$ and replacing $k$ by $k + 1$ produces a $90$ -element subset with sum $S + 1.$

Hence $S$ takes every value from $4095$ to $4995,$ for $4995 - 4095 + 1 = 901$ possible values.

2006 AIME I Problem 2

Solution:

Problem 2 in Other Years