= Solution
For $m\geq2$, $f''(x)=m(m-1)x^{m-2}\geq0$ on $[0,\infty)$, so $f$ is a <convex function>; for $m=1$ it is linear and also convex. The equally weighted <random variable> has <expected value> $EX=N^{-1}\sum_i x_i=1/N$. Applying <Jensen's inequality> gives
$$
\frac1N\sum_{i=1}^N f(x_i)=Ef(X)\geq f(EX)=f(1/N).
$$
Multiplying by $N$ proves
$$
\boxed{\sum_{i=1}^N x_i^m\geq N^{1-m}.}
$$
For $m>1$ equality requires all $x_i=1/N$, by strict convexity; for $m=1$ equality holds for every admissible vector.
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