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\sum a_i = 100 and \sum a_i^2 = 1000 , max a_i =?

Source: Switzerland - Swiss TST 2000 p2

February 19, 2020
inequalitiesSummaxalgebra

Problem Statement

Real numbers a1,a2,...,a16a_1,a_2,...,a_{16} satisfy the conditions ∑i=116ai=100\sum_{i=1}^{16}a_i = 100 and ∑i=116ai2=1000\sum_{i=1}^{16}a_i^2 = 1000 . What is the greatest possible value of a16a_16?
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