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Omni-MATH / Find the smallest positive number λ, such that for any 12 points on the plane P₁,P₂,…,P₁₂(can overlap), if the distance between any two of them does not exceed 1, then …

Problem

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problem

Find the smallest positive number

\lambda

, such that for any

12

points on the plane

P_1,P_2,\ldots,P_{12}

(can overlap), if the distance between any two of them does not exceed

1

, then

\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda

Plain-text mathematical notation (without MathML)

Find the smallest positive number λ, such that for any 12 points on the plane P₁,P₂,…,P₁₂(can overlap), if the distance between any two of them does not exceed 1, then ∑_(1≤i<j≤12)|P_(i)P_(j)|²≤λ.

Original LaTeX notation

Find the smallest positive number $\lambda$, such that for any $12$ points on the plane $P_1,P_2,\ldots,P_{12}$(can overlap), if the distance between any two of them does not exceed $1$, then $\sum_{1\le i<j\le 12} |P_iP_j|^2\le \lambda$.

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task_id: 66f2b029-0341-51e6-8b31-977b458c8375
task_key: test--66f2b029-0341-51e6-8b31-977b458c8375
task_revision_id: 3
config / split: / test

Official source

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