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Omni-MATH / Let n≥4 be an integer. Find all functions W : {1, dots, n}^2 to mathbb R such that for every partition [n]=A∪B∪C into disjoint sets, ∑_(a∈A)∑_(b∈B)∑_(c∈C)W(a,b)W(b,c)=|A||B||C|.

Problem

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problem

Let

n \ge 4

be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition

[n] = A \cup B \cup C

into disjoint sets,

\sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|.

Plain-text mathematical notation (without MathML)

Let n≥4 be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition [n]=A∪B∪C into disjoint sets, ∑_(a∈A)∑_(b∈B)∑_(c∈C)W(a,b)W(b,c)=|A||B||C|.

Original LaTeX notation

Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, \[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. \]

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task_id: 4bc775ac-9e46-5a6f-8313-435a0cdb3730
task_key: test--4bc775ac-9e46-5a6f-8313-435a0cdb3730
task_revision_id: 3
config / split: / test

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