HMMT 十一月 2020 · THM 赛 · 第 5 题

HMMT November 2020 — THM Round — Problem 5

The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let N be the original number of the room, and let M be the room number as shown on the sign. The smallest interval containing M a c all possible values of can be expressed as [ , ) where a, b, c, d are positive integers with gcd( a, b ) = gcd( c, d ) = 1. N b d Compute 1000 a + 100 b + 10 c + d .

解析

The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let N be the original number of the room, and let M be the room number as shown on the sign. M a c The smallest interval containing all possible values of can be expressed as [ , ) where a, b, c, d are N b d positive integers with gcd( a, b ) = gcd( c, d ) = 1. Compute 1000 a + 100 b + 10 c + d . Proposed by: Andrew Lin Answer: 2031 Solution: Let A represent the portion of N to the right of the deleted zero, and B represent the rest of N . For example, if the unique zero in N = 12034 is removed, then A = 34 and B = 12000. Then, A + B/ 10 M 9 B = = 1 − . N A + B 10 N k The maximum value for B/N is 1, which is achieved when A = 0. Also, if the 0 removed is in the 10 ’s k k +1 place ( k = 2 in the example above), we find that A < 10 and B ≥ 10 , meaning that A/B < 1 / 10 and thus B/N > 10 / 11. Also, B/N can get arbitrarily close to 10 / 11 via a number like 1099 . . . 9. M 1 2 Therefore the fraction achieves a minimum at and always stays below , though it can get N 10 11 1 2 arbitrarily close. The desired interval is then [ , ). 10 11