HMMT 十一月 2023 · THM 赛 · 第 8 题

HMMT November 2023 — THM Round — Problem 8

Call a number feared if it contains the digits 13 as a contiguous substring and fearless otherwise. (For example, 132 is feared, while 123 is fearless.) Compute the smallest positive integer n such that there exists a positive integer a < 100 such that n and n + 10 a are fearless while n + a, n + 2 a, . . . , n + 9 a are all feared.

解析

Call a number feared if it contains the digits 13 as a contiguous substring and fearless otherwise. (For example, 132 is feared, while 123 is fearless.) Compute the smallest positive integer n such that there exists a positive integer a < 100 such that n and n + 10 a are fearless while n + a, n + 2 a, . . . , n + 9 a are all feared. Proposed by: Rishabh Das Answer: 1287 Solution: First of all, note that we cannot have n, n + a, . . . , n +10 a be less than 1000, since we cannot have fearless numbers have 13 as their last two digits since a < 100, and 129 , 130 , 131 , . . . , 139 doesn’t work as 139 is feared. Thus, we must utilize numbers of the form 13 xy , where 1, 3, x , and y are digits. If all of n + a, n + 2 a, . . . , n + 9 a start with 13, then a ≤ 12, and the minimum we can achieve is 1288, with 1288 , 1300 , 1312 , . . . , 1384 , 1396 , 1408 . If, however, n + 9 a = 1413, then we can take a = 14 to get 1287 , 1301 , 1315 , . . . , 1399 , 1413 , 1427 , so the minimum possible value is 1287.