NMC 1990

Let $m, n$, and $p$ be odd positive integers. Prove that the number

$$\sum_{k=1}^{(n-1)^{p}} k^{m}$$

is divisible by $n$.

Let $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers. Prove

$$\sqrt[3]{a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3}} \leq \sqrt{a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}}$$

When does equality hold in (1)?

Let $A B C$ be a triangle and let $P$ be an interior point of $A B C$. We assume that a line $l$, which passes through $P$, but not through $A$, intersects $A B$ and $A C$ (or their extensions over $B$ or $C$ ) at $Q$ and $R$, respectively. Find $l$ such that the perimeter of the triangle $A Q R$ is as small as possible.

It is possible to perform three operations $f, g$, and $h$ for positive integers: $f(n)=$ $10 n, g(n)=10 n+4$, and $h(2 n)=n$; in other words, one may write 0 or 4 in the end of the number and one may divide an even number by 2. Prove: every positive integer can be constructed starting from 4 and performing a finite number of the operations $f, g$, and $h$ in some order.