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A fairly inelegant inequality

Source: Germany 2012 - Problem 4

December 5, 2022
inequalitiesalgebrasuquare rootreal number

Problem Statement

Let a,ba,b be positive real numbers and n≥2n\geq 2 a positive integer. Prove that if xn≤ax+bx^n \leq ax+b holds for a positive real number xx, then it also satisfies the inequality x<2an−1+2bn.x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.
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