HMMT 二月 2023 · 冲刺赛 · 第 2 题

HMMT February 2023 — Guts Round — Problem 2

[10] Let n be a positive integer, and let s be the sum of the digits of the base-four representation of 2 − 1. If s = 2023 (in base ten), compute n (in base ten). ◦

解析

[10] Let n be a positive integer, and let s be the sum of the digits of the base-four representation of n 2 − 1. If s = 2023 (in base ten), compute n (in base ten). Proposed by: Dongyao Jiang Answer: 1349 Solution: Every power of 2 is either represented in base 4 as 100 . . . 00 or 200 .. 00 with some number 4 4 n of zeros. That means every positive integer in the form 2 − 1 is either represented in base 4 as 333 . . . 33 or 133 . . . 33 for some number threes. Note that 2023 = 2022 + 1 = 674 · 3 + 1, meaning 4 n 2 − 1 must be 133 . . . 333 with 674 threes. Converting this to base 2 results in 4 674 1349 133 . . . 33 = 200 . . . 00 − 1 = 2 · 4 − 1 = 2 − 1 4 4 for an answer of 1349. ◦