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Inequality with cubed reciprocals

Source: Czech and Slovak Olympiad 1972, National Round, Problem 1

July 10, 2024
inequalitiesalgebra

Problem Statement

Show that the inequality ∏k=2n(1−1k3)>12\prod_{k=2}^n\left(1-\frac{1}{k^3}\right)>\frac12 holds for every positive integer n>1.n>1.
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