$1 \cdot 75$ and $\sqrt[3]{5+2 \sqrt{13}}+\sqrt[3]{5-2 \sqrt{13}}$ equal to
(A) $\frac{3}{2}$.
(B) $\frac{\sqrt[3]{65}}{4}$.
(C) $\frac{1+\sqrt[6]{13}}{2}$.
(D) $\sqrt[3]{2}$
(E) None of these.
(31st American High School Mathematics Examination, 1980)
[Solution] Let $a=\sqrt[3]{5+2 \sqrt{13}}, b=\sqrt[3]{5-2 \sqrt{13}}$, and $x=a+b$.
Then $\quad x^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$,
which means
$$
x^{3}=a^{3}+b^{3}+3 a b(a+b),
$$

Substituting the given values into the equation, we get $x^{3}=10+3 \sqrt[3]{-27} x$,
or
$$
(x-1)\left(x^{2}+x+10\right)=0,
$$

which has only the real solution $x=1$. Therefore, $a+b=1$.
Thus, the answer is $(E)$.