HMMT 十一月 2024 · 冲刺赛 · 第 6 题

HMMT November 2024 — Guts Round — Problem 6

[6] The vertices of a cube are labeled with the integers 1 through 8 , with each used exactly once. Let s be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of s over all such labelings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2024, November 09, 2024 — GUTS ROUND Organization Team Team ID#

解析

[6] The vertices of a cube are labeled with the integers 1 through 8, with each used exactly once. Let s be the maximum sum of the labels of two edge-adjacent vertices. Compute the minimum possible value of s over all such labelings. Proposed by: Derek Liu Answer: 11 Solution: The answer must be at least 11, because the label 8 is adjacent to three vertices, one of which has label at least 3. To show 11 is achievable, note that the following labelling achieves s = 11: 8 1 2 7 3 6 5 4 Thus the answer is 11 .