HMMT 十一月 2008 · GEN1 赛 · 第 6 题

英文解析

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 aresparse.
   √ √ √
   3 5 7
   Answer: -1,0,1 For any number x , other than these 3, x, x, x, x, . . . provide infinitely manypossible values of s , so these are the only possible sparse numbers. On the other hand, − 1 is the onlypossible 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.