HMMT 十一月 2008 · GEN1 赛 · 第 6 题

HMMT November 2008 — GEN1 Round — Problem 6

[ 5 ] We say “ s grows to r ” if there exists some integer n > 0 such that s = r . Call a real number r “sparse” if there are only finitely many real numbers s that grow to r . Find all real numbers that are sparse.

解析

[ 5 ] We say “ s grows to r ” if there exists some integer n > 0 such that s = r . Call a real number r “sparse” if there are only finitely many real numbers s that grow to r . Find all real numbers that are sparse. √ √ √ 3 5 7 Answer: -1,0,1 For any number x , other than these 3, x, x, x, x, . . . provide infinitely many possible values of s , so these are the only possible sparse numbers. On the other hand, − 1 is the only possible value of s for r = − 1, 0 is the only value for r = 0, and − 1 and 1 are the only values for r = 1. Therefore, − 1, 0, and 1 are all sparse.