Competition problem — Combinatorics

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Combinatorics Difficulty 5.0 AIME, harder Prove it United States

Problem:

Shelly writes down a vector $v=(a, b, c, d)$ , where $0<a<b<c<d$ are integers. Let $\sigma(v)$ denote the set of 24 vectors whose coordinates are $a, b, c$ , and $d$ in some order. For instance, $\sigma(v)$ contains $(b, c, d, a)$ . Shelly notes that there are 3 vectors in $\sigma(v)$ whose sum is of the form $(s, s, s, s)$ for some $s$ . What is the smallest possible value of $d$ ?

Solution

Solution:

If $k = a + b + c + d$ , first you notice $4 \mid 3k$ , and $k \geq 10$ . So we try $k = 12$ , which works with $a, b, c, d = 1, 2, 3, 6$ and not $1, 2, 4, 5$ .

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