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sum a_ia_j(1 - a_ia_j) >= 0 if sum a_1 = sum a_i^2, for a_i>0

Source: 2015 Saudi Arabia IMO TST II p3

July 24, 2020
inequalitiesalgebran-variable inequality

Problem Statement

Let a1,a2,...,ana_1, a_2, ...,a_n be positive real numbers such that a1+a2+...+an=a12+a22+...+an2a_1 + a_2 + ... + a_n = a_1^2 + a_2^2 + ... + a_n^2 Prove that ∑1≤i<j≤naiaj(1−aiaj)≥0\sum_{1\le i<j\le n} a_ia_j(1 - a_ia_j) \ge 0 Võ Quốc Bá Cẩn.
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