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\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}+ ... indepedent of a

Source: Spanish Mathematical Olympiad 1990 P4

August 2, 2018

algebraradical

Problem Statement

Prove that the sum

\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}

is independent of

a

for

a \ge - \frac{3}{4}

and evaluate it.