2015 IMO Shortlist N8

For every positive integer $n$ with prime factorization $n=\prod_{i=1}^{k} p_{i}^{\alpha_{i}}$, define $$\mho(n)=\sum_{i: p_{i}>10^{100}} \alpha_{i} .$$ That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that $$\mho(f(a)-f(b)) \leq \mho(a-b) \quad \text { for all integers } a \text { and } b \text { with } a>b .$$

对于每个具有质因数分解 $n=\prod_{i=1}^{k} p_{i}^{\alpha_{i}}$ 的正整数 $n$，定义 $$\mho(n)=\sum_{i: p_{i}>10^{100}} \alpha_{i} 。$$即$\mho(n)$是$n$大于$10^{100}$的质因数的个数，按重数计算。查找所有严格递增函数 $f: \mathbb{Z} \rightarrow \mathbb{Z}$ 使得 $$\mho(f(a)-f(b)) \leq \mho(a-b) \quad \text { 对于所有整数 } a \text { 和 } b \text { 且 } a>b 。$$