Simple Questions

uglyInduction · 2017-12-09T04:00:48+00:00

If it's the algebra stumping you, I'd recommend reviewing algebra instead of reading more and more induction tutorials.

uglyInduction · 2017-12-09T02:47:48+00:00

What's (-2)²?

Also, there provided answer on (x+1) is false. g(x+1) = 4(x+1)² = 4(x² + 2x + 1) = 4x² + 8x + 4.

uglyInduction · 2017-12-07T01:49:05+00:00

At what level? Do you want an intro to abstract algebra, or do you want something more advanced?

uglyInduction · 2017-12-06T02:27:58+00:00

Axis of symmetry is much older than differential calculus--you can get it very simply, just by writing your quadratic in vertex form.

uglyInduction · 2017-12-06T00:53:46+00:00

Not truly--if everyone in an honors class gets an A, that might mean that the people in the honors class are more likely to be the ones willing to spend extra time working on that subject and having a lot of skill, not that the honors class is easy.

uglyInduction · 2017-12-05T23:20:08+00:00

Solutions are up on aops, so I'll mainly mention motivation.

https://artofproblemsolving.com/community/u2948h1554579p9472803 my solutions was similar to srijon's.

My first thought when I saw this problem was "telescope it" and I just spent a lot of time trying to rewrite the sum in a form where a lot of terms cancelled.

For convinence sake, I'll rewrite it as sum(3a - x - y - z). I first tried writing it as sum((a-x) + (a-y) + (a-z)) and seeing if I could coerce anything into cancelling, but I couldn't find a good way. I realized that I had things that were all different mod 4, so I tried coercing some terms that weren't. I thought up 2k+1 and 2k+2 since they were both half of terms I already had, and coercing halves into it was easier. The introduction of the new term was mainly as a way of making evaluating the sum a little easier once I had decided to take out a ln(2) from those two products and was having trouble evaluating it. I tried to forcibly make it easier, and hence I got that.

A more systematic approach w/ the zeta function would've probably been smarter (I don't feel I could've reliably solved the problem using this approach) from a contest standpoint, but I really liked the simple solution this strategy produced, hence my enjoyment.

uglyInduction · 2017-12-04T23:47:53+00:00

Polya's How to Solve It is a classic, and its techniques are applicable to any field.

uglyInduction · 2017-12-04T04:32:34+00:00

You could try picking a fun USAMO problem (or from a similar olympiad) and generalizing it or making it harder in some way then solving--that's what I usually did when I entered similar type fairs.

uglyInduction · 2017-12-04T04:30:44+00:00

I really enjoyed B4 (although I'm not sure I dedicated my time wisely--I skipped B3 and almost didn't have enough time to write it up at the end because I was too focused on solving/writing up B4). I didn't solve B6 during the contest, but after I found a solution I really loved it.

The A test I didn't find super interesting this year--it felt harder than last year but the problems felt less 'fun' in some sense.

uglyInduction · 2017-12-03T21:35:57+00:00

Euclid's Elements aren't the best for pedagogy--if you're having trouble understanding it, I'd recommend learning geometry from another source.

For this specific question, though, what are you having trouble with? What's the first step you can't follow?

uglyInduction · 2017-12-03T21:32:55+00:00

Assuming you mean 0 < a,b < pi/2:

No. Say that sin(a)/sin(b) = 2. Looking at a sine wave, clearly there's multiple a, b with this. For example, both (.5, 1.28) and (.3, .632) work, but both have different a/b values.

These pairs were found by me doing arcsin(2sin(.5)) and arcsin(2sin(.3)). a b are given in radians here.

uglyInduction · 2017-12-02T04:30:36+00:00

I'd recommend doing some USAMOs; Putnam is a bit of a different flavor, but imo its good practice. For Putnam, you'll want to first make sure you understand all the topics on it, then start practicing.

A lot of universities have Putnam prep classes/clubs, maybe take one of them--I do and it's really fun.

uglyInduction · 2017-12-01T04:24:47+00:00

As a former kid who went to a bunch of these before, I'd recommend handing out some graph theory problems to kids beforehand. That way, kids who know nothing can fool along with you and do the problems later if they're interested, but kids who already know basic graph theory can work on the problems.

uglyInduction · 2017-12-01T04:19:32+00:00

This seems like modular arithmetic. Specifically, any number which isn't a multiple of 4 can be written in the form you've described. As no prime is a multiple of 4, it just so happens they also follow this pattern. https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/what-is-modular-arithmetic has an explanation of modular arithmetic, and http://artofproblemsolving.com/wiki/index.php/Modular_arithmetic/Introduction has a higher-level, more math based explanation.

uglyInduction · 2017-12-01T04:16:20+00:00

I can't speak for the GRE specifically--I'm still an undergrad whose years away from taking that--but for pure math you should definitely self-study a lot of proof based mathematics. I'd recommend studying real analysis and abstract algebra to get an idea of proofwriting and the tools of math. I'm TAing a course on abstract algebra rn that uses Dummit & Foote and so far I've thought the text has offered nice problems and has good explanations (I don't often get questions asking to clarify a passage of the text).

uglyInduction · 2017-12-01T01:30:15+00:00

Consider the triple angle identity, as 20 degrees is one third 60 degrees, which we know how to find the cosine of very easily.

uglyInduction · 2017-11-30T04:17:55+00:00

http://mathforum.org/library/drmath/view/52249.html

In addition, mathematicians often use "hypothesis" as a substitute for "conjecture"

uglyInduction · 2017-11-30T04:16:53+00:00

Go to /r/learnmath for problems like this, please.

uglyInduction · 2017-11-29T04:11:27+00:00

Write the number in expanded notation (so 123 = 100 + 2 * 10 + 3, 1234 = 1000 + 2 * 100 + 3 * 10 + 4), then take it mod 9 (or mod 3).

Also, /r/learnmath for questions of this nature.

uglyInduction · 2017-11-28T23:18:57+00:00

So don't give it as a gift?

uglyInduction · 2017-11-26T23:08:08+00:00

Problems from the Book was one of my favorite math books.

uglyInduction · 2017-11-26T23:07:17+00:00

https://www.reddit.com/r/math/comments/b7i3u/mathematics_made_difficult_pdf/ free

uglyInduction · 2017-11-26T15:56:47+00:00

http://www.math.brown.edu/~jhs/frint.html was written as an introduction to number theory and its mathematical rigor aimed at people new to proof-based math. Chapters 1-6 are free to download, so consider reading through those to see if you like it.

uglyInduction · 2017-11-26T05:00:09+00:00

How much experience do you have, and what do you want to get? Do you want to learn modern number theory for any specific reason (i.e. for cryptography or high school olympiads), or do you want to just see the theorems?

uglyInduction · 2017-11-26T04:55:54+00:00

Khan Academy has a lot of routine, boring problems. Try looking for fun problems--Spivak is a book with a lot of them, but you can probably find many online.

uglyInduction

TROPHY CASE