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Smaller than [n(n+1)/2]^2

Source: Austrian Federal Competition 2013, part 2, problem 4

June 18, 2013
inequalitiesfunctioninductioninequalities proposed

Problem Statement

For a positive integer nn, let a1,a2,…ana_1, a_2, \ldots a_n be nonnegative real numbers such that for all real numbers x1>x2>…>xn>0x_1>x_2>\ldots>x_n>0 with x1+x2+…+xn<1x_1+x_2+\ldots+x_n<1, the inequality ∑k=1nakxk3<1\sum_{k=1}^na_kx_k^3<1 holds. Show that na1+(n−1)a2+…+(n−j+1)aj+…+an⩽n2(n+1)24.na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.
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