HMMT 十一月 2022 · 冲刺赛 · 第 7 题

HMMT November 2022 — Guts Round — Problem 7

[7] All positive integers whose binary representations (excluding leading zeroes) have at least as many 1’s as 0’s are put in increasing order. Compute the number of digits in the binary representation of the 200th number.

解析

[7] All positive integers whose binary representations (excluding leading zeroes) have at least as many 1’s as 0’s are put in increasing order. Compute the number of digits in the binary representation of the 200th number. Proposed by: Isabella Quan Answer: 9 Solution: We do a rough estimation. There are 255 positive integers with at most 8 digits and a majority of them, but not more than 200, satisfy the property. Meanwhile, there are 511 positive integers with at most 9 digits, and a majority of them satisfy this property. Thus, the answer must be greater than 8 and at most 9.