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Romanian District Olympiad 2000, Grade IX, Problem 1

Source:

September 24, 2018
inequalitiesromaniaInequalityalgebra

Problem Statement

a) Show that n2≥2x+3x3+⋯+nxnn−1−x, \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, for all non-negative reals x x and integers n≥2. n\ge 2.
b) If x,y,z∈(0,∞), x,y,z\in (0,\infty ) , then prove the inequality ∑cycx(2x+y+z)2+4≤3/16 \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16
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