Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME Prove it Soviet Union

Problem:
Can we label each vertex of a 45-gon with one of the digits $0, 1, \ldots, 9$ so that for each pair of distinct digits $i, j$ one of the 45 sides has vertices labeled $i, j$ ?

Solution

Solution:
$10 \times 5 > 45$ , so some digit $i_0$ must appear less than $5$ times. But each occurrence can give at most $2$ edges $i_0, j$ , so there are at most $8$ edges $i_0, j$ , which is one too few.

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