  Mathematical reasoning is an important capability of large language
models~(LLMs) for real-world applications. To enhance this capability, existing
work either collects large-scale math-related texts for pre-training, or relies
on stronger LLMs (\eg GPT-4) to synthesize massive math problems. Both types of
work generally lead to large costs in training or synthesis. To reduce the
cost, based on open-source available texts, we propose an efficient way that
trains a small LLM for math problem synthesis, to efficiently generate
sufficient high-quality pre-training data. To achieve it, we create a dataset
using GPT-4 to distill its data synthesis capability into the small LLM.
Concretely, we craft a set of prompts based on human education stages to guide
GPT-4, to synthesize problems covering diverse math knowledge and difficulty
levels. Besides, we adopt the gradient-based influence estimation method to
select the most valuable math-related texts. The both are fed into GPT-4 for
creating the knowledge distillation dataset to train the small LLM. We leverage
it to synthesize 6 million math problems for pre-training our JiuZhang3.0
model, which only needs to invoke GPT-4 API 9.3k times and pre-train on 4.6B
data. Experimental results have shown that JiuZhang3.0 achieves
state-of-the-art performance on several mathematical reasoning datasets, under
both natural language reasoning and tool manipulation settings. Our code and
data will be publicly released in
\url{https://github.com/RUCAIBox/JiuZhang3.0}.
