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Sum of 1/\sqrt{x^3+1} when sum of 1/x=3

Source: Tuymaada 2014, Day 1, Problem 4 Juniors, Problem 3 Seniors

July 12, 2014
inequalitiesfunctionTuymaada

Problem Statement

Positive numbers a, b, ca,\ b,\ c satisfy 1a+1b+1c=3\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3. Prove the inequality 1a3+1+1b3+1+1c3+1≤32.\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}.
(N. Alexandrov)
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