NMC 1988

The positive integer $n$ has the following property: if the three last digits of $n$ are removed, the number $\sqrt[3]{n}$ remains. Find $n$.

Let $a, b$, and $c$ be non-zero real numbers and let $a \geq b \geq c$. Prove the inequality

$$\frac{a^{3}-c^{3}}{3} \geq a b c\left(\frac{a-b}{c}+\frac{b-c}{a}\right)$$

When does equality hold?

Two concentric spheres have radii $r$ and $R, r<R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $A B C$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \leq 2 r$.

Let $m_{n}$ be the smallest value of the function

$$f_{n}(x)=\sum_{k=0}^{2 n} x^{k}$$

Show that $m_{n} \rightarrow \frac{1}{2}$, as $n \rightarrow \infty$.