Question

You must spend as much time as you can _________(study)hard at English to catch up with others.

Answer 1

You must spend as much time as you can____（study）hard at English to catch up with others.你必须花费尽可能的时间去努力学习英语赶上其他人。
can后面省去了spend，所以study是动词不定式作目的状语，故应用to study。

Answer 2

You must spend as much time as you can _________(study)hard at English to catch up with others.
这个题是一个典型的【两难选择】，即用两种形式都有道理。也就是说，这是个【失败的题】。
1、You must spend as much time as you can【 studying】hard at English to catch up with others.
这就是尽人皆知的【spend time doing sth】结构,花费时间做某事。
2、You must spend as much time as you can 【to study】hard at English to catch up with others.
这就是传说中的【do as much as one can to do sth】尽某人的最大努力来做某事。
祝你开心如意！