HMMT 二月 2024 · 冲刺赛 · 第 3 题

HMMT February 2024 — Guts Round — Problem 3

[5] Compute the number of even positive integers n ≤ 2024 such that 1 , 2 , . . . , n can be split into pairs, 2 and the sum of the numbers in each pair is a multiple of 3 .

解析

[5] Compute the number of even positive integers n ≤ 2024 such that 1 , 2 , . . . , n can be split into 2 pairs, and the sum of the numbers in each pair is a multiple of 3 . Proposed by: Rishabh Das Answer: 675 Solution: There have to be an even number of multiples of 3 at most n , so this means that n ≡ 0 , 2 (mod 6) . (We can also say that there should be an equal number of 1 (mod 3) and 2 (mod 3) numbers, which gives the same restriction.) We claim that all these work. We know there are an even number of multiples of 3 , so we can pair them; then we can pair 3 k + 1 and 3 k + 2 for all k . 2022 This means the answer is + 1 = 675 . 3