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In my past 3 years learning mathematics, and by that I mean high school standard and above, I have encountered numerous moments of highs and lows. Highs when you managed to prove a difficult theorem or find a rare integration result, and lows when you are simply stumped by a difficult Olympiad question.

Throughout these experiences, I came up with a few quotes that really stuck to me, quotes that carried me through the journey. I wish to share these quotes with you and hope to provide an insight in problem solving or simply the finer aspects of the subject.

1. To find x, all you need is a little faith.
As I moved further into high level mathematics, the problems I faced usually have multiple approaches to solve. Some approaches are more tedious then others but not all approaches lead to the correct answer. And then there are problems where I get stuck in a particular approach and have given up, only to find that approach to be the only way to the solution. At times like these, I remind myself to have that little faith because amidst the horrendous calculations, the answer might just be around the corner.

2. I've done nothing if it's not proven for all x.
A quote regarding proofs. Many times, I may find a special case for which the theorem is proven true. When some students find such special cases, which isn't quite easy in itself, they tend to feel pleased that their hard work has paid of. But is that what the theorem is, to prove for some special case. Apparently not. I spend no time to enjoying such special cases. I need to prove for all x.

3. Just think about it long enough, your mind will give you the answer.
A important quote my English tuition teacher gave me. I guess the keyword here is 'will'. It is questionable whether you will definitely get the answer if you though about it long enough. Nevertheless, it does make you persevere when working with very difficult problems.

4. Have a plan. And then execute it.
Simply put: Always have a plan. Be it in life or in mathematics, I create for myself a list of steps that I need to take to get anything done. For my webpage, I plan out the modules I will conduct. For my band, I mark out the list of songs I need to work on. For my examinations, I divide the material I need to study for each day. For mathematics, I chart out my route to learn a certain concept. While planning does help, it will amount to nothing if you don't execute it.

5. How long do I have to solve this problem?
Time is of the essence. I have grown up to be a person that doesn't like to waste time and more importantly, doesn't like to waste other people's time. When someone approaches me to solve a problem, be it mathematics or otherwise, I usually ask the person 'how much time do I have?'. While I may not get the correct answer, working with a time frame in mind does ensure I get the best solution I can given the time I have.

6. All right, I'm alive again. Let's go!
It is no surprise that one finds many difficulties in Olympiad questions. This is particularly true when the student attempts questions on a topic that requires a set of theorems in another topic altogether, think doing a probability question using some calculus. When I moved to solving Putnam problems, I was overwhelmed with the myriad of 'cross topic' questions that at times I wanted to give up. This quote then encouraged me. I picture myself as a gun slinger. During gun fights, he may run out of rounds, just like how I exhaust my list of techniques and inequalities to solve the problem. But just like the gun slinger, I tell myself to take a tea break, reload on more techniques, and then face the problem again. Let's go!

7. You got the answer now. You mind checking the domain of x.
When I started learning mathematics, I always overlooked the conditions and terms imposed on certain equations. All of us are familiar with the 'cannot divide by 0' rule. In high level mathematics, we need to pay more careful attention to other things like domains, ranges, inclusive, exclusive, continuous, derivatives exists, etc. Do not run the risk of finding a solution to a problem only to find that it is invalid due to a violation on a certain condition.

8. It's no big deal. There are more important matters at hand.
It seems that the people I interact each day spend plenty of time talking about pointless matters and don't focus on the problem at hand. Take for example, organizing an 'integration bee'. I walk inside a room only to find people shunning responsibilities, delegating jobs, assigning teams, and the most common proving that their idea is better than the rest. Rarely will there be that person who simply stands up and say, "Tell me how many problems you need and I'll think of them." Relating to mathematics, I try to be that kind of person. When working on a conjecture, I throw out every irrelevant aspect that gets in the way. Focus on the conjecture and something great may come out of it.

9. Unlike history, there is only one right answer.
This is one of the main reasons why I like mathematics. I consider it to be a perfect science. In my other arts subjects, I also hear teachers say, "What do you think the answer is. Come one, there is no right or wrong answer." To some, they like finding their own answers and substantiating it. To me, I like giving an answer where there is zero room for error.

10. Put me in prison and I will prove the Riemann hypothesis.
This quote came right after reading a book on the Riemann hypothesis. Doing high level math requires the utmost concentration of the mind. While I do agree that an aspect of progress is through discussion with fellow mathematicians, there are also moments of enlightenment which only happens when the mathematician gives his full undivided focus at the problem at hand. He needs to be in a place void of people, hobbies and other distractions, so that each train of thought is filled with nothing but numbers. I have never reached such a state before. While, perhaps because I've never been in prison.