how to approach a math competition

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mmmcannibalism
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how to approach a math competition

Postby mmmcannibalism » Sun Mar 07, 2010 2:55 am UTC

Not sure how familiar everyone will be, but my areas regional math day is in a few weeks and I am having trouble deciding how to prepare(really want to qualify for state hopefully with a first place). My main problem in deciding approach is that its hard to know what types of problems will be on the test.

From past experience, there are a lot of in depth word problems along with a lot of applied algebra 2 and pre calculus type questions. My main problem is the problems that rely on specific types of knowledge that can't be worked around by general math skill, for instance having to know that if a number has a remainder of 1 when divided by 7 then that number to a whole number power will also have a remainder of 1. I'm hoping since xkcd is a glorious hive of knowledge and mathstuffs some of you guys/gals know of any particular site that would be useful in learning various types of word problems and interesting math properties(like the divide by 7 thing)

tldr; can anyone help me find a site to study for a math contest, focus on obscure parts of algebra 2/pre calc and weird quirky math.
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cpp789
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Re: how to approach a math competition

Postby cpp789 » Sun Mar 07, 2010 3:33 am UTC

Actually that "weird" thing works with all moduli, not just 7. It comes from the theory of congruences, which is a branch of number theory. I'm not sure of any good sources for learning that, but wikipedia is always a good place to start.

Also, if you want to compete in math contests, check out the Noah sheets. http://staff.imsa.edu/math/journal/volume4/articles/NoahSheets.pdf

Good luck.

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Re: how to approach a math competition

Postby achan1058 » Sun Mar 07, 2010 4:00 am UTC

It really depends on the style of the contest. The thing that always works is to look at old contests, obviously. Aside from that, it really depends on whether it is a straight forward calculation contest, an applied calculation contest (ie. if you try to do the obvious thing, it won't work, so you need to find some trick), or a proof contest. For calculation types, you would need to be able to make calculations accurately, for obvious reasons. For proof types, good write up is essential. As for specific knowledge, I would say elementary number theory tends to be popular in contests in general, but it really depends on the people setting it.

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Re: how to approach a math competition

Postby Qaanol » Sun Mar 07, 2010 5:46 am UTC

The three most important things: get plenty of sleep the night before, eat a good breakfast, and don't stress out.
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DavCrav
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Re: how to approach a math competition

Postby DavCrav » Sun Mar 07, 2010 11:34 am UTC

mmmcannibalism wrote:...My main problem is the problems that rely on specific types of knowledge that can't be worked around by general math skill, for instance having to know that if a number has a remainder of 1 when divided by 7 then that number to a whole number power will also have a remainder of 1....tldr; can anyone help me find a site to study for a math contest, focus on obscure parts of algebra 2/pre calc and weird quirky math.


The cheese burnt me a bit at seeing the entire branch of modular arithmetic being described as weird and quirky. My experience has been that these mathematics competitions (the ones that test ability rather than normal exams) can be made a lot easier if you know: basic modular arithmetic; airthmetic-geometric mean; some crazy geometry theorems. As the last is quite difficult to learn (there are lots) go for AMGM and modular arithmetic.

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Re: how to approach a math competition

Postby duckshirt » Sun Mar 07, 2010 4:24 pm UTC

One specific thing that's often useful on math competitions that I've taken is the Euler characteristic, stating that V - E + F = 2 for a 3-d polyhedron with no 'holes' (where V is the number of vertices, E is the number of edges, F is the number of faces). If there are holes, you can figure out other values it will equal (e.g. if there's one hole, it's topologically equivalent to the torus, and V - E + F = 0). It's something that, in my opinion, is easy to remember but isn't obvious to derive on the fly.
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mmmcannibalism
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Re: how to approach a math competition

Postby mmmcannibalism » Sun Mar 07, 2010 5:29 pm UTC

DavCrav wrote:
mmmcannibalism wrote:...My main problem is the problems that rely on specific types of knowledge that can't be worked around by general math skill, for instance having to know that if a number has a remainder of 1 when divided by 7 then that number to a whole number power will also have a remainder of 1....tldr; can anyone help me find a site to study for a math contest, focus on obscure parts of algebra 2/pre calc and weird quirky math.


The cheese burnt me a bit at seeing the entire branch of modular arithmetic being described as weird and quirky. My experience has been that these mathematics competitions (the ones that test ability rather than normal exams) can be made a lot easier if you know: basic modular arithmetic; airthmetic-geometric mean; some crazy geometry theorems. As the last is quite difficult to learn (there are lots) go for AMGM and modular arithmetic.


Thanks for the advice on those, I think the fact that someone could be annoyed on behalf of math is the reason this site is so useful for this type of thing.

Thanks for the help so far everyone, ima start looking at the modular arithmetic and geometry noah sheets(which look amazing).
Izawwlgood wrote:I for one would happily live on an island as a fuzzy seal-human.

Oregonaut wrote:Damn fetuses and their terroist plots.

DavCrav
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Re: how to approach a math competition

Postby DavCrav » Mon Mar 08, 2010 9:44 am UTC

mmmcannibalism wrote:
DavCrav wrote:
mmmcannibalism wrote:...My main problem is the problems that rely on specific types of knowledge that can't be worked around by general math skill, for instance having to know that if a number has a remainder of 1 when divided by 7 then that number to a whole number power will also have a remainder of 1....tldr; can anyone help me find a site to study for a math contest, focus on obscure parts of algebra 2/pre calc and weird quirky math.


The cheese burnt me a bit at seeing the entire branch of modular arithmetic being described as weird and quirky. My experience has been that these mathematics competitions (the ones that test ability rather than normal exams) can be made a lot easier if you know: basic modular arithmetic; airthmetic-geometric mean; some crazy geometry theorems. As the last is quite difficult to learn (there are lots) go for AMGM and modular arithmetic.


Thanks for the advice on those, I think the fact that someone could be annoyed on behalf of math is the reason this site is so useful for this type of thing.

Thanks for the help so far everyone, ima start looking at the modular arithmetic and geometry noah sheets(which look amazing).


I wasn't annoyed. I lol'd... (I might have got confused but I remember lol used to change into 'the cheese is burning me'...)

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Re: how to approach a math competition

Postby skeptical scientist » Mon Mar 08, 2010 3:32 pm UTC

DavCrav wrote:I wasn't annoyed. I lol'd... (I might have got confused but I remember lol used to change into 'the cheese is burning me'...)

old language filter wrote:¡This cheese is burning me!
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Re: how to approach a math competition

Postby Yakk » Mon Mar 08, 2010 4:25 pm UTC

The Binomial theorem (pascal's triangle, etc) is a good thing to know. Learning it might teach you useful basic combinatorics as well.

Doing mathematics using logarithms can be useful -- one HS level competition I used to write always did something involving large powers of the year, for which logarithmic mathematics tended to be key.

That 7 trick, for example, can be derived from the binomial thorem.

(a+b)^k = a^k + a^(k-1)*b * (k choose k -1) + ... + a^1 * b^(k-1) * (k choose k-1) + b^k

Suppose your number x has a remainder of 1 when divided by 7. Then x = m7 + 1 for some value 7.

(m7+1)^k = (m7)^k + (m7)^(k-1)*1 * ... + (m7)*1^(k-1) * (k choose k-1) + 1^k
Everything except the last term has a multiple of 7 in it. So, there is a value n such that
(m7+1)^k = 7 * n + 1
which means that when you divide (m7+1)^k by 7, you get n with remainder 1.

If you do this with modular arithmetic, the entire problem becomes:
x == 1 mod 7
thus
x^k == 1^k mod 7
thus
x^k == 1 mod 7
which is much shorter and easier. ;-)
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.


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