HMMT 二月 2020 · 冲刺赛 · 第 4 题

HMMT February 2020 — Guts Round — Problem 4

[4] Compute the value of 105 − 104 , given that it is a positive integer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2020, February 15, 2020 — GUTS ROUND Organization Team Team ID#

解析

[4] Compute the value of 105 − 104 , given that it is a positive integer. Proposed by: Andrew Gu Answer: 181 3 3 2 2 Solution 1: First compute 105 − 104 = 105 + 105 · 104 + 104 = 3 · 105 · 104 + 1 = 32761. Note 2 2 2 that 180 = 32400, so 181 = 180 + 2 · 180 + 1 = 32761 as desired. 3 3 2 2 Solution 2: We have 105 − 104 = 105 + 105 · 104 + 104 . Thus √ √ √ 3 3 104 3 < 105 − 104 < 105 3 . √ √ Now estimating gives the bounds 180 < 104 3 and 105 3 < 182. So the answer is 181.