HMMT 十一月 2023 · 冲刺赛 · 第 15 题

HMMT November 2023 — Guts Round — Problem 15

[9] Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2023, November 11, 2023 — GUTS ROUND Organization Team Team ID#

解析

[9] Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23, only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the board. Proposed by: Andrew Wen Answer: 321 Solution: Let the original numbers be m < n . We know mn = ( m − 20)( n + 23) = mn − 20 n + 23 m − 460 = ⇒ 23 m − 20 n = 460 . Furthermore, 23 m < 23 n hence 460 < 3 n = ⇒ n ≥ 154. Furthermore, we must have 23 | n hence the least possible value of n is 161 which corresponds m = 160. This yields a minimum sum of 161 + 160 = 321.