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E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2}

Source: Moldova TST 2023

April 9, 2023
inequalities

Problem Statement

Let n n (n≥2)(n\geq2) be an integer. Find the greatest possible value of the expression E=a11+a12+a21+a22+…+an1+an2E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2} if the positive real numbers a1,a2,…,ana_1,a_2,\ldots,a_n satisfy a1+a2+…+an=n2.a_1+a_2+\ldots+a_n=\frac{n}{2}. What are the values of a1,a2,…,ana_1,a_2,\ldots,a_n when the greatest value is achieved?
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