<p>While I can do some AIME problems, ace Collegeboard math exams without a sweat, and self-study BC Calculus in barely any time at all, I can't do many of the simplest of proof-based textbook problems, to my immense frustration. I also thought that I could pursue no more mathematics. It was also a major frustration for me during SIMUW, where I was definitely able to understand non-proof-based material more than proof-based material. I am inspired of the example of another SIMUW student, who got 6 on his AIME and was also lost when it came to proofs. Proofs were so mysterious to me that out of all possible outcomes, I couldn't wait for the classes to end, whenever they covered proofs! It can be explained, perhaps, by my lack of training in proofs. But when I think of other people who have acquired understanding of proofs prematurely, I have to ask, where in the heck did they get their skills? </p>
<p>The question is, where and when do most of you AoPSers and talented math students acquire knowledge of proof methods? And is it a huge struggle up to the point in which it finally gets into your subconsciousness, as D. Solow explained? Most math competitions up to the level of AIME do not use the proofs. Nor are they taught in math classes, beyond the unmathematical two column proof of geometry. </p>
<p>I'm 16 years old now. D. Solow explains that proofs are best learned by 8th grade (one has to wonder how people get that knowledge). I do not know if it is too late. But I am nonetheless inspired by one particular person I knew, who went from not considering college to becoming a college mathematics superstar. It may take an entire read of D. Solow's entire book.</p>
<p>I think it is a very deep system of thinking, much like doing research. I am not big on competitoins, but from my research experience, I have seen that it takes tremendous insight and a little luck to hit upon a solution. It isn't easy stuff, and for every one person that gets it, 10 will fail.</p>
<p>I like proofs because it is the definition of absolute certainty.</p>
<p>Hey, I'm sixteen, too. It's impossible for me to know exactly how you feel, but I can definitely relate to the feeling of, "Where do these guys learn this stuff?"</p>
<p>You never learn proofs in regular high school. I wasted one year of my life by not dedicating myself to learning geometry the real way. My class moaned and groaned about how hard proofs were, so after second nine-weeks, my teacher assigned no more proofs.</p>
<p>From my experience, I believe that the best students in the world acquire their math olympiad training by some of the following methods:</p>
<ol>
<li><p>They are the children of parents who are familiar with olympiad style questions. Instant premium grade resource.</p></li>
<li><p>They are in contact with other students who are relatively strong in math olympiads and learn from them. These students are very self-dedicated and motivated.</p></li>
<li><p>In the case of China, it's selected for. The best students are singled out and trained nationally by the country's best high school teachers to perform at peak level.</p></li>
</ol>
<p>Many people don't know it, but geometry is a lot harder than it is when taught properly. Trig at the high school level is pure memorization. Calculus is definitely hard and requires a lot of hard work, but when you take the AP exam, it's whether or not you are familiar with the rules and know how to use them. I honestly don't think that I could come up with a proof of Euler's Line or the construction of a regular heptadecagon by myself. I think if these problems were introduced at the high school level many students would think differently of geometry.</p>
<p>"I like proofs because it is the definition of absolute certainty."</p>
<p>Isn't a mathematic proof only certain and absolutely true if and only if you assume the quantized nature of the things you are trying to explain? Isn't modern mathematics based on "quantum" thinking?</p>
<p>How can you say that proofs are certain if no one can yet prove the nature of the universe?</p>
<p>I learned proofs last year (junior year) in my regular math class. We also have Integrated Math (III), which is dumb and different from every other high school's math system.</p>
<p>I was really confused at PROMYS when we were doing number theory proofs. Yet, everyone else seemed to know what was going on. I have never had prior experience with any proofs before though. They are pretty facinating nonetheless.</p>
<p>Isn't a mathematic proof only certain and absolutely true if and only if you assume the quantized nature of the things you are trying to explain? Isn't modern mathematics based on "quantum" thinking?</p>
<ul>
<li>A good proof does that. Of course, you can never say that about science, because we can't make those assumptions. Since we created mathematics essentially, we can prove things with certainty.</li>
</ul>
<p>Of course, that is assuming you have a real proof, not a flawed proof.</p>