4

Part of 2001 Irish Math Olympiad

Problems(2)

nice inequality

Source: Ireland 2001

Prove that for all positive integers

n

: \frac{2n}{3n\plus{}1} \le \displaystyle\sum_{k\equal{}n\plus{}1}^{2n}\frac{1}{k} \le \frac{3n\plus{}1}{4(n\plus{}1)}.

inequalitiesinequalities proposed

nonnegative real numbers

Source: Ireland 2001

Find all nonnegative real numbers

x

for which \sqrt[3]{13\plus{}\sqrt{x}}\plus{}\sqrt[3]{13\minus{}\sqrt{x}} is an integer.

algebra proposedalgebra