I'm currently prepping for some high school math competitions soon, and I was wondering if anyone knows any resources that are out there with an abundance of contest-math-related geometry problems. Geometry is definitely my weak point in contest math, and any input would be appreciated. Thanks!
[Math] Contest Math Geometry
contest-mathgeometry
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Added: On the pragmatic side of things, participation in structured extra-curricular activities is a "good thing" to be able to note in your future applications to college. In that sense, if you are not presently engaged in extra-curricular activities (for lack of interest, say, in sports, etc), but you like math, participating in math competitions would we a good match. If you are interested in pursuing a degree in math (or math-related field), such participation/dedication also shows that your interest in math, in particular, extends beyond the classroom.
That said:
Learning math isn't just about learning "things": content, definitions, rules, theorems, etc. That is, it isn't just about "what you learn or know" but also
- math is about "how you learn, how you think, how you solve problems, and how you use the "what" of what you're learning"
- and math is about the "why" and how to demonstrate that you know "how to..." (e.g. solve problems, construct proofs), and "why" (justifying, fully understanding why things work)...
So preparing for math competitions, to the extent that doing so makes you a more agile thinker, develops creativity, intuition, problem solving strategies, developing facility with proofs and logic, etc., will all help you in the long run, in math, and in life. Some of the "tricks" you may pick up and many of the things you simply memorize may not stick, and likely will not help much, in the long run. But the "activity" of doing math and engaging with challenging problems will transform you, your mind and your brain.
[Personally, I like to think of "mathematics" as a verb, as well as a noun!]
So, the short answer: Yes, preparing for math competitions will be useful - in that you will benefit, and it is relevant in that you will learn, and learn how to learn, you will learn how to better perform under time constraints, you will have the opportunity to engage with others who love doing math and meet new people, particularly if your "eye is [not only] on the prize.
Besides, if it's fun, and you like what you're doing, isn't that reason enough?
Your question on mental abilities is of course too vague to admit a definitive answer, but I'll try to give some reflections on the subject.
1) Essentially, I strongly believe that the differences in abilities necessary to tackle the different branches of mathematics are vastly exaggerated.
In my experience good mathematicians are good at any subject.
The difference between their choices results from mathematics having become so vast that it is very difficult or impossible to have expertise in several subjects, unless you are Serre or Tao.
But my conviction, formed by introspection and anecdotal evidence, is that the subject mathematicians end up with very much depends on chance: books found in a library when 16 years old, teachers had in high-school or university, admired friends,...
To be quite honest, some subjects like combinatorics seem to require special gifts and be a little isolated, but even that is changing: I'm thinking of combinatorists like Stanley who use quite sophisticated "mainstream" mathematics, commutative algebra for example.
2) As for algebraic geometry, it certainly requires no special gifts.
Its origin is Descartes's (and Fermat's) fantastic invention of coordinate geometry, which allows one to solve difficult geometric problems by algebra, in an essentially purely mechanical way (which by the way Jean-Jacques Rousseau didn't like: read the extract from his Confessions in the epigraph to Fulton's Algebraic Curves, page iii)
In the 19-th century and in the 20-th century up to about the 1960's hard algebraic geometry was taught (under the name "analytic geometry") in high schools and a Swiss friend of mine showed me problems he solved when 17 years old, which would baffle most Ph.D holders in algebraic geometry nowadays.
The problem is that, in order in particular to solve quite classical problems and also for arithmetic reasons, algebraic geometers like van der Waerden, Zariski, Weil, Serre, Grothendieck,... had to introduce quite sophisticated machinery, culminating in the notion of scheme.
The unfortunate consequence of those developments is that too many introductory courses spend a semester (say) setting up this wonderful modern machinery and have no time left for showing how to apply it to concrete problems.
The good news is that an antidote to this state of affairs exists: it is called math.stackexchange !
I am amazed at the quality, concreteness and pertinence of many questions and answers relating to algebraic geometry here, and I can only advise you to become a frequent user of our site: just use the tag [(algebraic-geometry)] and start reading, pen in hand!
Best Answer
A good place to start is http://www.artofproblemsolving.com which covers all kinds of contest math.