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Nice Algebra

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September 16, 2023
algebra

Problem Statement

Given n∈N+,n≥3,a1,a2,⋯ ,an∈R+.n\in\mathbb N_+,n\ge 3,a_1,a_2,\cdots ,a_n\in\mathbb R_+. Let b1,b2,⋯ ,bn∈R+b_1,b_2,\cdots ,b_n\in\mathbb R_+ satisfy that for ∀k∈{1,2,⋯ ,n},\forall k\in\{1,2,\cdots ,n\}, ∑i,j∈{1,2,⋯,n}\{k}i≠jaibj=0.\sum_{\substack{i,j\in\{1,2,\cdots ,n\}\backslash \{k\}\\i\neq j}}a_ib_j=0. Prove that b1=b2=⋯=bn=0.b_1=b_2=\cdots =b_n=0.
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