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Omni-MATH / Shelly writes down a vector v=(a,b,c,d), where 0<a<b<c<d are integers. Let σ(v) denote the set of 24 vectors whose coordinates are a,b,c, and d in some order. For instance, σ(v) co…

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problem

Shelly writes down a vector v=(a,b,c,d)v=(a, b, c, d), where 0<a<b<c<d0<a<b<c<d are integers. Let σ(v)\sigma(v) denote the set of 24 vectors whose coordinates are a,b,ca, b, c, and dd in some order. For instance, σ(v)\sigma(v) contains (b,c,d,a)(b, c, d, a). Shelly notes that there are 3 vectors in σ(v)\sigma(v) whose sum is of the form (s,s,s,s)(s, s, s, s) for some ss. What is the smallest possible value of dd?
Plain-text mathematical notation (without MathML)
Shelly writes down a vector v=(a,b,c,d), where 0<a<b<c<d are integers. Let σ(v) denote the set of 24 vectors whose coordinates are a,b,c, and d in some order. For instance, σ(v) contains (b,c,d,a). Shelly notes that there are 3 vectors in σ(v) whose sum is of the form (s,s,s,s) for some s. What is the smallest possible value of d?
Original LaTeX notation
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 $d$?

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Source and history

task_id
ac8522ed-7c8b-5f15-93f5-7d56742546fd
task_key
test--ac8522ed-7c8b-5f15-93f5-7d56742546fd
task_revision_id
1
config / split
/ test

Official source

initial import