HMMT 二月 2025 · 冲刺赛 · 第 36 题
HMMT February 2025 — Guts Round — Problem 36
题目详情
- [20] Ethan initially writes some numbers on a blackboard, each of which is either a 3 or a 5. He then repeatedly picks two numbers and replaces them with their sum, difference, product, or quotient (if the divisor is nonzero). Let f ( n ) denote the minimum number of numbers Ethan must initially write for him to be able to eventually write the number n . For example, f (2025) ≤ 6 because Ethan could start with 3, 3, 3, 3, 5, and 5 on the board, then repeatedly multiply two numbers at a time to eventually get 2025. Submit a comma-separated ordered 8-tuple of integers corresponding to the values of f (164), f (187), f (191), f (224), f (255), f (286), f (374), and f (479), in that order, or an X for any value you wish to leave blank. For instance, if you think f (164) = 9 and f (224) = 8, you should submit “9, X, X, 8, X, X, X, j k 2 ( C +1) W X”. You will earn 0 . 6 · points, where C is the number of correct answers you submit and W 4 is the number of incorrect (non-blank) answers.
解析
- [20] Ethan initially writes some numbers on a blackboard, each of which is either a 3 or a 5. He then repeatedly picks two numbers and replaces them with their sum, difference, product, or quotient (if the divisor is nonzero). Let f ( n ) denote the minimum number of numbers Ethan must initially write for him to be able to eventually write the number n . For example, f (2025) ≤ 6 because Ethan could start with 3, 3, 3, 3, 5, and 5 on the board, then repeatedly multiply two numbers at a time to eventually get 2025. Submit a comma-separated ordered 8-tuple of integers corresponding to the values of f (164), f (187), f (191), f (224), f (255), f (286), f (374), and f (479), in that order, or an X for any value you wish to leave blank. For instance, if you think f (164) = 9 and f (224) = 8, you should submit “9, X, X, 8, X, j k 2 ( C +1) W X, X, X”. You will earn 0 . 6 · points, where C is the number of correct answers you submit 4 and W is the number of incorrect (non-blank) answers. Proposed by: Derek Liu Answer: 6, 6, 6, 5, 5, 6, 6, 7 Solution: The following expressions represent optimal ways for Ethan to make each of the 8 given numbers. 164 = 3(5(5 + 5) + 3) + 5 187 = 3(3 + 5)(3 + 5) − 5 191 = 5 · 5(3 + 5) − 3 · 3 224 = (3 + 5)(5 · 5 + 3) 255 = 5 · 5(5 + 5) + 5 286 = (3 + 5 + 5)(5 · 5 − 3) 374 = 3 · 5 · 5 · 5 − 3 / 3 = 3 · 5 · 5 · 5 − 5 / 5 479 = (5 · 5 − 3)(5 · 5 − 3) − 5 = (3 − 5 · 5)(3 − 5 · 5) − 5 It can be checked by code that these are optimal.