HMMT November, 2024

Combinatorics Difficulty 5.0 AIME, harder Prove it United States

Problem:

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.

Solution

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 $s=11$ :

Thus the answer is $11$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.