HMMT 十一月 2025 · GEN 赛 · 第 2 题

HMMT November 2025 — GEN Round — Problem 2

Suppose n integers are placed in a circle such that each of the following conditions is satisfied: • at least one of the integers is 0; • each pair of adjacent integers differs by exactly 1; and • the sum of the integers is exactly 24. Compute the smallest value of n for which this is possible.

解析

Suppose n integers are placed in a circle such that each of the following conditions is satisfied: • at least one of the integers is 0; • each pair of adjacent integers differs by exactly 1; and • the sum of the integers is exactly 24. Compute the smallest value of n for which this is possible. Proposed by: Derek Liu Answer: 12 Solution: Note n is always even because the numbers in the circle alternate parity. If n ≤ 8, then the sum is at most 0 + 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16, so n ≥ 10. If n = 10, the sum consists of 5 even and 5 odd numbers, so it cannot be even. Thus, n ≥ 12 , achieved with the following diagram. 0 1 1 2 0 3 1 4 2 3 3 4