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Putnam 2013 A5

Source:

December 9, 2013
Putnamgeometryinequalitiessymmetryvectorintegrationgeometric transformation

Problem Statement

For m≥3,m\ge 3, a list of (m3)\binom m3 real numbers aijka_{ijk} (1≤i<j<k≤m)(1\le i<j<k\le m) is said to be area definite for Rn\mathbb{R}^n if the inequality ∑1≤i<j<k≤maijk⋅Area(△AiAjAk)≥0\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0 holds for every choice of mm points A1,…,AmA_1,\dots,A_m in Rn.\mathbb{R}^n. For example, the list of four numbers a123=a124=a134=1,a234=−1a_{123}=a_{124}=a_{134}=1, a_{234}=-1 is area definite for R2.\mathbb{R}^2. Prove that if a list of (m3)\binom m3 numbers is area definite for R2,\mathbb{R}^2, then it is area definite for R3.\mathbb{R}^3.
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