HMMT 十一月 2022 · 冲刺赛 · 第 31 题

HMMT November 2022 — Guts Round — Problem 31

[17] Given positive integers a , a , . . . , a such that 1 2 2023 2023 X a = | a − a | k k i i =1 for all 1 ≤ k ≤ 2023, find the minimum possible value of a + a + · · · + a . 1 2 2023

解析

[17] Given positive integers a , a , . . . , a such that 1 2 2023 2023 X a = | a − a | k k i i =1 for all 1 ≤ k ≤ 2023, find the minimum possible value of a + a + · · · + a . 1 2 2023 Proposed by: Maxim Li Answer: 2046264 Solution: Without loss of generality, let a ≤ a ≤ · · · ≤ a . Then, note that 1 2 2023 2023 X a − a = | a − a | − | a − a | k +1 k k +1 i k i i =1 = k ( a − a ) − (2023 − k )( a − a ) k +1 k k +1 k = (2 k − 2023)( a − a ) . k +1 k Thus, a = a unless k = 1012, so a = a = · · · = a and a = · · · = a , and we can check k +1 k 1 2 1012 1013 2023 that that they must be in a ratio of 1011 : 1012. Thus, a , . . . , a must consist of 1012 copies of 1 2023 1011 c , and 1011 copies of 1012 c for some c , so for everything to be a positive integer, we need c ≥ 1. This gives us the final answer of 1011 · 1012 + 1012 · 1011 = 2046264.