TITLE: (Soft) What maths should I concentrate on at 16-18 years old?
QUESTION [11 upvotes]: Some background information first of all:
I'm 16 now and just started studying mathematics intensely. I live in the UK and my goal is to eventually become very good at advanced mathematics (graduate level and up) but that is still far in the future. I'm about a year or slightly more ahead from the expected curriculum knowledge in my age group.
The advice I'm seeking of is that I am not sure whether to keep going ahead of the curriculum and eventually learning university maths a year before actually going to uni or should I try to improve my problem solving skills by for example trying out Mathematical Olympiad questions. If I do MO instead then I fear I might be wasting time (even if I'm improving at problem solving) when all I really want to do is the more advanced stuff which is more proof laden I hear anyway.
To answer the question it might be helpful to know what my intentions for pursuing mathematics are. I enjoy learning new fields of mathematics that I did not know of before. A new topic excites me, a new problem to solve with the same tools just doesn't have the same effect on me mentally. I can pick up stuff quiet quickly and get bored if I'm forced to keep doing what I already know without any new knowledge. I realise however that mathematical ability is not just about knowledge but also problem solving skill.

REPLY [3 votes]: I would recommend reading Velleman's How to Prove it, which teaches you about how to think logically and construct mathematical proofs. This is often the biggest hurdle for new mathematics students: figuring out what constitutes a proof, what are logically valid arguments, and how to present a proof. 
Alongside Velleman, I would recommend Axler's Linear Algebra Done Right. Linear algebra is the study of vector spaces (and later on, modules) and the linear transformations between them. Linear algebra plays a key role in every field of mathematics, as well as in physics and computer science. For example, the derivative operation in calculus is all about linearizing functions.