2006 IMO Shortlist N3

The sequence $f(1), f(2), f(3), \ldots$ is defined by $$f(n)=\frac{1}{n}\left(\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor\right),$$ where $\lfloor x\rfloor$ denotes the integer part of $x$. (a) Prove that $f(n+1)>f(n)$ infinitely often. (b) Prove that $f(n+1)<f(n)$ infinitely often. (South Africa)

序列 $f(1), f(2), f(3), \ldots$ 定义为 $$f(n)=\frac{1}{n}\left(\left\lfloor\frac{n}{1}\right\rfloor+\left\lfloor\frac{n}{2}\right\rfloor+\cdots+\left\lfloor\frac{n}{n}\right\rfloor\right),$$ 其中$\lfloor x\rfloor$ 表示$x$ 的整数部分。 (a) 证明$f(n+1)>f(n)$无限次出现。 (b) 无限次证明 $f(n+1)<f(n)$。 (南非)