AT-OEMO-FED1 1991

Suppose that $a,b,$ and $\sqrt[3]{a}+\sqrt[3]{b}$ are rational numbers. Prove that $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are also rational.

Solve in real numbers the equation:

$\frac{1}{x}+\frac{1}{x+2}-\frac{1}{x+4}-\frac{1}{x+6}-\frac{1}{x+8}-\frac{1}{x+10}+\frac{1}{x+12}+\frac{1}{x+14}=0.$

Find the number of squares in the sequence given by $a_0=91$ and $a_{n+1}=10a_n+(-1)^n$ for $n \ge 0.$

Let $AB$ be a chord of a circle $k$ of radius $r$ , with $AB=c$ . $(a)$ Construct the triangle $ABC$ with $C$ on $k$ in which a median from $A$ or $B$ is of a given length $d.$ $(b)$ For which $c$ and $d$ is this triangle unique?