aand 
Prove that
is 
Find the sum of the digits of 
. Prove that
on a plane with centers 
,
is feet of the perpendicular from 
to 
and 
and 
are intersections of line 
with circle 
.
and 
can move around the circle 
and 
,
.
an even number.
is not the product of consecutive integers for infinitely many naturals 
.
,
.
be the point on 
such that 
.
.
,
are relatively prime positive integers and 
is a prime number, determine the value of 
.
![\begin{align*}
[(a+1)x-(b-1)y][ax+by]\\
\end{align*}](http://latex.artofproblemsolving.com/b/5/2/b52d93a771b06b5b0a0bebc986de07eda44f25f4.png)
is a polynomial function such that 
then
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Disclaimer: This entire post is my opinion, and note that my perspective blinds me from certain things, so I would appreciate any feedback.
Contest math is pretty much 33% attitude, 33% hard work, 33% luck, and 1% talent.
Basically, a lot of people actually work a lot to get good at math and don't actually learn much compared to someone who just looks at a math problem once a week or so. Probably most extreme example of this is people at high schools who study a few hours/day for math by rereading the text over and over again and doing random exercises from textbooks. That's.... a rather inefficient way of learning math.
So, do hard problems. There's more to approaching math though.
This is a pretty amazing quote, because not only does it say the secret to doing well on contest math much shorter than I could ever express it in, it also doesn't mention that this is actually very helpful to doing well.
I'll try to explain why its so powerful, though reading CatalystOfNostalgia's post above probably will be much more helpful.
Treat the problems you're doing as interesting places you want to explore. Just... try to figure out everything, look at how interesting it is, see the ridiculous connections that make it tricky. Try to understand it. Don't just do a problem and forget it, the problem should be interesting, you should remember it forever, you should seek to understand it, etc...
For this, and also hard work, I recommend working with other people. It's pretty useful.
Not much you can do about having bad luck and getting a USAMO with 6 geometry problems or something, so just try to not have that happen.
Like... it's easy to think btw that you're doing everything that you could, but you don't have enough talent or some random thing like that. It's usually because you're not doing everything that you could, see if you REALLY are working on problems how you should be working.
Doing math actually takes up a lot of energy, at least for me. Makes me hungry pretty easily.
Other stuff that helps:
Books:
standard is Aopsv1->aopsv2->some aops introduction series->Art and Craft of Problem Solving by Paul Zeitz->Problem Solving Strategies->Whatever you want for olympiad level. There are a ton of good olympiad books out there, any of them should work.
Also, read some books about cool math. Math is pretty cool.
Problem Sources:
preparing for AMC/AIME is usually done best with actual AMCs/AIMEs.
Olympiad level, I highly recommend russian olympiads. other than that IMO Shortlist is pretty ok i guess, though some problems are rather... bad. China TSTs can work. So can USAMOs.
If you're just starting, getting any book is usually a pretty decent start and what I would recommend.
Finally, add math people on gmail. (Assuming they aren't super-busy and have no time to talk to you.) Being able to reach other math people and talk about math problems is pretty useful. Also they can give you pretty good problems.
Point out anything that could be written better in this post, I've historically been pretty bad at writing explanations/opinions/things in general.
Contest math is pretty much 33% attitude, 33% hard work, 33% luck, and 1% talent.
Basically, a lot of people actually work a lot to get good at math and don't actually learn much compared to someone who just looks at a math problem once a week or so. Probably most extreme example of this is people at high schools who study a few hours/day for math by rereading the text over and over again and doing random exercises from textbooks. That's.... a rather inefficient way of learning math.
So, do hard problems. There's more to approaching math though.
CatalystOfNostalgia wrote:
I guess what this boils down to is that you need to appreciate the process of reaching your goals at least as much as reaching them. Whether or not you reach them shouldn't affect your appreciation in the end for what you've learned, the skills you've developed. Don't work because you want to reach a goal; work because you enjoy the work you're doing.
This is a pretty amazing quote, because not only does it say the secret to doing well on contest math much shorter than I could ever express it in, it also doesn't mention that this is actually very helpful to doing well.
I'll try to explain why its so powerful, though reading CatalystOfNostalgia's post above probably will be much more helpful.
Treat the problems you're doing as interesting places you want to explore. Just... try to figure out everything, look at how interesting it is, see the ridiculous connections that make it tricky. Try to understand it. Don't just do a problem and forget it, the problem should be interesting, you should remember it forever, you should seek to understand it, etc...
For this, and also hard work, I recommend working with other people. It's pretty useful.
Not much you can do about having bad luck and getting a USAMO with 6 geometry problems or something, so just try to not have that happen.
Like... it's easy to think btw that you're doing everything that you could, but you don't have enough talent or some random thing like that. It's usually because you're not doing everything that you could, see if you REALLY are working on problems how you should be working.
Doing math actually takes up a lot of energy, at least for me. Makes me hungry pretty easily.
Other stuff that helps:
Books:
standard is Aopsv1->aopsv2->some aops introduction series->Art and Craft of Problem Solving by Paul Zeitz->Problem Solving Strategies->Whatever you want for olympiad level. There are a ton of good olympiad books out there, any of them should work.
Also, read some books about cool math. Math is pretty cool.
Problem Sources:
preparing for AMC/AIME is usually done best with actual AMCs/AIMEs.
Olympiad level, I highly recommend russian olympiads. other than that IMO Shortlist is pretty ok i guess, though some problems are rather... bad. China TSTs can work. So can USAMOs.
If you're just starting, getting any book is usually a pretty decent start and what I would recommend.
Finally, add math people on gmail. (Assuming they aren't super-busy and have no time to talk to you.) Being able to reach other math people and talk about math problems is pretty useful. Also they can give you pretty good problems.
Point out anything that could be written better in this post, I've historically been pretty bad at writing explanations/opinions/things in general.
and add your friends from there.
Someone give me something that fits it.
Starcraft.
