I have created a diagram of most contests and programs for high school students in Suffolk County who have interest in mathematics and/or computer science. It might be helpful for non-Suffolk students, beware some information are location sensitive.
This small list was made so no one have to walk my way anymore. I had no idea there are contest I could join until 11th grade was half over. Many of the contests require school's participation, you will have to talk to the principle (he handles the money..) and your math/computer science teacher.

These competitions are opens to high school students and most likely do-able for Suffolk students. Like there is no way a Suffolk student is going to take Rice University Mathematics Tournament without getting funds and sitting on a bus(car) for days1. All contest requires traveling are either within 200 mile radius centered in Suffolk, or there are organized trips that cut down the cost. Only contests with actual value prize or enough publicity are listed, there are a lot problem of the week/month contests, there is no way to mention them all.
The difficulty means the difficulty of individual question(ignore the time limit) or in other cases(like Intel Science Talent Search), the difficulty of gaining a high position in the contest.
Arrows means one have to do well in one contest to be accepted to join the contest arrow points to.
For more detailed information on each contest, Wikipedia it(recommended for more famous contests, usually their site sucks.) or follow the links to the official site.

Mathematics competitions
The most common ones suffolkian can encounter
The easiest math teasers: Colstate Question of The Week(Columbus State University Math Contest)
Features 4 questions every week at Monday 0:00(not exactly, usually before 0:20). Two of them are for high school students. Problem of the week and Algebra in Action. An correct entry will be chosen randomly for each contest and will be awarded with a what's your problem t-shirt.(2 chance per week)
The question are usually extremely easy, the only fun part about this simple contest would be become the first one to solve it. Person who answers the problem correctly will be listed on their site.
I wait till 0 am every time, the list of people who solved it are ordered chronically.

The most common local math contest on Suffolk: Math League.
I believe a lot of schools(at least 14) in Suffolk has a math teams compete with other schools though math league. This is like the no.1 easiest way to compete with other math students in real life. There are 6 problems, each 2 problems have a 12 minute time restriction. Problems are fairly easy and usually have an algebraic answer.

NYSML: New York State Mathematics League, require some registration fee. Usually schools will pay for you. To be selected, you have to make into Suffolk County Math Team[SCMT, unofficial abb.]. I believe it's determined by performance on math league. Local math league coaches can also nominate 2 students with the qualified students. Contest method and questions similar to ARML.

ARML: American Regions Mathematics League. Like NYSML, you have to make into the SCMT. There are a few sites for the contest, Pennsylvania State University is the site for SCMT. There will be 4 different rounds:

1. a individual round, with 10 minute time limit on 2 problems, total of 8 problems and 40 minutes.
2. A team round of 20 minutes to solve 10 problems.
3. Relay, 3 people of a team work on 3 questions one by one
4. Power, proof based problem for the entire team to solve in an hour

Suffolk County Mathematics Tournament: Sponsored by Suffolk County Math Teachers Association and taking place in Suffolk County Community College. Each school can bring at most 4 teams to the contest. The fees usually are covered by individual schools instead of the students. The coach select the teams. Consist of 3 rounds and calculators are allowed in each one of them.

1. 15 questions, each grade get different one. 45 minutes to solve all 15 questions, usually have algebraic solutions.
2. 10 multiple choice questions with 5 choices. No grade distinctions.I believe it has a 30 or 25 minute time limit.2 This and the above count as individual score
3. 16 team questions for 45 minutes(I think). I believe team score counts some individual scores.

The winner will be announced shortly after the end of the team round.

The road to IMO
The most prestigious mathematic contest in the world for high school student is the International Mathematical Olympiad. There are a few paths to achieve that in US, but both have to go though AIME and USAMO.

AMC: The American Mathematics Competitions is one of the ways to get into AIME. 25 questions in 75 minutes. The top student of AMC for 4 years = one gold medal(maybe not real gold). The top 1% of AMC 10 and top 5% of AMC 12 goes into AIME. AIME have around 15000 people each year, if we assume same amount of people take AMC 10 and 12. then around half million people take AMC each year. Around 18000 people took it in NY in 2008.

USAMTS: USA Mathematical Talent Search. 5 question per month. Yeah, use any thing you want and submit your answer... average 6 days per questions, you should have perfect amount of time to do research and finish them. Asking for help from other people to solve it for you would be cheating. The proofs gets your points, the thoroughness of the proof will determine how much point you can get(max is 5). People with most points at end of the year can get a few books and other goodies. Top 30 male and female will be invited into AIME.

AIME: American Invitational Mathematics Examination. The name says clearly it's invitational. Only around 15000 people join it each year, 2006 has 22000...-.- There will be 15 questions with 3 hour time limit, the questions are hard, mode are usually 2 or 3 out of 15. Answers are integers between 0 to 999, a lucky guess gives you a winning chance.
Top 500 goes into USAMO.

USAMO: Be one of the participant of USA Mathematical Olympiad is my ultimate goal next year. It shows I'm one of the top 500 people in mathematics (currently, I'm assume I'm around the top 20000 in US). It's proof based, like IMO, except a bit easier. Below are taken directly from the official site.

The USAMO is a six question, two day, 9 hour essay/proof examination. All problems can be solved with pre-calculus methods. Approximately 500 of the top scoring AMC participants (based on a weighted average) are invited to take the USAMO. U. S. citizens and students legally residing in the United States and Canada (with qualifyng scores) are eligible to take the USAMO.

12 people will be selected to go to Washington D.C. for awards, and they will be invited into MOSP.

MOSP:
Mathematical Olympiad Summer Program is the summer program preparing for students to represent US in IMO. Out of 12 people, only 6 will be chosen. I heard it's very intense. Some notes taken from MSOP by Reid Barton can be found here.

IMO
International Mathematical Olympiad, 6 people can make up a team for a country. 9 hours in 2 days for 6 questions. It's said the 4th question are always the easiest one. There is a lot info on IMO in every language, there was even a book written about it --Count Down: The Race for Beautiful Solutions at the International Mathematical Olympiad
Gold in IMO is the Fields medal for high school students.

Other Math contests
HMMT
The Harvard-MIT Mathematics Tournament is a very difficult challenge with years the best scorer get below 50%. The award is a Klein bottle. Great way to visit a college if your future goal is MIT(obviously, because you are reading this post) or Harvard. Students can form teams with students from other schools.

PUMaC
Princeton Math Competition. I don't know much about it, but seems like the same difficulty level as HMMT.

USAMTS alike
These contests are in USAMTS format, 1 month and solve a few questions and send to the grader online.
Wisconsin Mathematics Talent Search
The best scorer can compete for $24000 scholarship(for University of Wisconsin only). Do it if that's your fav school. and Honor day, but seems like for Wisconsin residents only.

iTest
Another USAMTS alike. Prize is $200 gift card(for 2007). But seriously... all that work for $200 only seems like a bit unworthy. I suggest it's best to put time for other contests unless you are really free.

Contests like the Math League
These are contests administrated like math league, in one's own school and mail in the result. Great scorers qualify for awards. I know very little about them because sadly, their official site is not user friendly at all. These are Log 1(one page description and... yep that's all there is to it), Rocket City Math League(seriously... why use flash for navigation?), The Mandelbrot Competition.

Mathematical Modeling
M3 challenge and HiMCM are the top 2. Here is what I learned from the M3 challenge. I don't like applied math, read about HiMCM yourself. But M3 challenge really beat most pure math contest... prize money is... wow!!! $20000 for the winning team!!! Applied math does earn more than pure math.
My suggestion, don't take mathematical modeling because you are good with pure math, it really doesn't help much.

Computer Science competitions
St. Joseph's Annual High School Programming Contest in Long Island
My school(Shoreham Wading-river High School, SWR) participates it every year. The winning team member gets $500 each. Each time can consist from 1 to 3 people. Language allowed: C++, Java and VB. Half Hollow Hills East and Smithtown dominated the last 5 years. This year SWR will take the goal3.

USACO
Language allowed: C, C++, Java, and Pascal.
USACO have 4 parts
The USACO training program. Here, do small training problems.
Six Internet competitions, it's like the AIME for CS.
US Open, another competition similar to Internet competition, it's like the USAMO for CS.
Training camp for IOI, similar to MOSP.

IOI
The IMO of informatics(or computer science). Each country can have 4 participants. It last 2 days. Since people join computer science competition are far fewer than mathematic competitions, IOI is a lot easier than IMO, personally speaking. IMO problems are impossible to me, IOI is at least do-able. Gold in IOI is the Turing Award for high school students.

Project Euler
This is in no way a competition(there is no awards...), but I have to mention this. There a many programming contests taken online in different fields, AI, e-commerce and other stuff, but Project Euler is the No.1 place to increase both mathematical and programming skills. It is filled with challenging mathematic oriented programming problems, a new one is added every week. After finish one problem, you can browse the forum for that particular problem, where participants show others how they did it. I just wish its server is more stable, it has way too much down time.

The math League alike in computer science
ACSL
Another math league like system, but instead of math, it's computer science. Top students can get free books.

GHOP: The Google Highly Open Participation Contest are made for high school students. Google allow students to work on any of the 10 open source projects. Finish one task = a GHOP t-shirt. Each time when 3 task is done, $100 will be awarded, max $500. The person does the best job in 1 open source project gets a chance to tour Googleplex.

There is no ACM High School competition sites around Long Island area.
Other competitions for computer science? yes, there are a lot, way too much to cover, I can only point out the most important ones and the most local ones for you. Most of the contests can be taken remotely and don't have much requirements like the math competitions do.

The general contest basically saying "we will reward you for anything scientific/mathematical/technological."
Intel Science Talent Search/Siemens Competition
These 2 competition have the best website among all the competition I have mentioned, just read their site and you will know why they are the most organized and most weighted competitions around. Also they offer the most money. Also become a grand finalist is as good as getting gold on IMO. Even Intel is said to be an science competition, it rewards math and computer science scholars like Siemens Competition. It's the best value one out there(and yep, take a lot of preparation). Become a finalist is the Nobel Prize for high school students.
The non contests
For computer science genius who finished college materials, take the GRE CS Subject test.
For math genius who finished college materials, take the GRE math Subject test.
Math summer camps: MathCamp, HCSSiM, PROMYS, MOSP if you are lucky. Ask the colleges in long island about summer programs, they sometimes have good stuff.
Math Socialites: Mu Alpha Theta, AMS(I applied for membership but never received a response), MAA.

The Master plan
Get active!

1. Make your school one of the Mu Alpha Theta chapter.
2. Make sure you can get a team of students who love math
3. Start to plan for funding
4. Print a copy of this page so you have a list of everything out there, find the dates of the competition, and register for them
5. it's actually possible to participate in every single thing I mentioned, there is no date conflict(at least in 2007-2008 period) except summer programs(if you are a freshman, then you have a chance to complete them all). But join 20 math contest... even I will call that as "having no life"

What are the ones most stand out to colleges?
The top 3: IMO, Intel, Siemens. Participate in IMO(or MOSP), Intel finalist, Siemens finalist. Almost means a free pass to any college of your choice. It's way better than a 2400 in SAT and 4.0 GPA
The nice ones: USAMO, Intel, Siemens, ARML, IOI. Participation in USAMO, IOI. Semi finalist in Intel, Siemens. One of the top individuals in ARML. These would put some weight to your college application. It's said that people participated in USAMO get into HYPS and MIT. But I know there are people who got into USAMO and getting rejected by a lot of schools due to poor GPA, SATs. Why is IOI not in the top? I personally feel all other Olympiads are much easier than IMO. Admission might like IMO a lot more than IOI. But there are only two IOI gold medalist from US last year, so it's more rare. Putting it here seems good enough.
Others:
Everything else doesn't really have as much weight as a 4.0GPA + 2400 SATs, maybe getting full credit on both GRE subject tests(that means your CS/math skill are beyond average first year graduate students) can create some good impressions. Doing well on HMMT and PUMac can definitely help if you are trying to get into their college.

Finally, this is 6 hours of work and some extra research before I started this article. Excuse my poor English. I hope you find it useful. I'm part of Suffolk County Math Team, I plan to help the community to do better academically in the future. I also hope you do fine on all competitions them, don't stress out over them. There is no way to improve one's math skill dramatically4, so do a lot of preparation when you still can(yes, I'm talking to you, freshman).

There is a great site to investigate for more:
Home School Math Contests

1. 1. unless you are super rich, of course, then you don't need this article. You can find a lot more on Wikipedia
2. 2. I can't remember, and the association should have a better website
3. 3. Very reasonable prediction
4. 4. Well there is always one way to improve skills dramatically, seal oneself in a room with all the math material one need and block all the possible escape routes. Food and drinks will be provided. Do that for 14 hours on weekends and 8 hours on normal days seems to create geniuses. I know someone did that for piano and now a musician. But, not recommended at all due to high risk of permanent psychological damage.

Sites devoted about sharing contests, like My Blog Contest and The Prize Blog, features more and more blog contests.
The usual blog contests features these few requirements:
1. Blog about the contest with specific anchor text
2. Blog about the blog holding the contest with specific anchor text
3. Comment or use some other form of communication to let the contest creator know

Usually, a blog contest means:
1. More visitors
2. Better ranking in search engines for some specific keywords
3. Some visitors might subscribe to the blog

A huge amount of benefit can be brought of the contest starter.
How about the contestants?
How about the Internet?

The contestants spend one post just to describe a contest to gain a small chance in winning some small price. If the post is unrelated to their blog, some readers of the contestant's blog will not be happy.

The Internet is created for information1. The blog contests is a way to play with the search engine. Contests are bribing bloggers to review their blog, like PayPerPost. Even a small blog with no much potential in some keyword but with high incentive prize, it can be increased in search engine rank. Thus the readers will read some poor quality blog because the blog paid some money for it.
PayPerPost users got punished by Google. Google will figure out a way of attack blog contests.

1. 1. I know the saying "Internet is for porn" but porn is information.

Rishi started a writing contest about "how I see my blog after one year from today".
The prize is $111, something I would like to work for, and it's less than half a month till the time I first started this blog (back then it was called webdevlogs).
Review the changes I had from last year, it's not difficult to imaging that my blog will have a completely new look a year later.
My life is filled with different changes. The ones that affect this blog most, is the change of interest. When the blog first started, it's about PHP, web development. Now, I occasionally post ideas on those field, most of the time I'm consumed by my other interests, like mathematics.Next year, my interest suppose to be the same thing, but less frequent updates due to the school work.
Zero spam comment like always, I don't give spam a 2nd chance.
Google adsense earning approaching the horizontal asymptote of $0, can't expect me to compete in the mathematic niche and win.
but, anyway, this blog will be a great show case for the colleges that consider me, wish the admission is not a grammar Nazi.
I guess, I would have 800+ post by that time, certainly some of them is to show off my project euler stats. [Why there is no chicks dig that?]
Some post will be my learning note on C++ and thoughts on game creation. I always dreamed to create a LF2 clone, but enable a lot player to interact online, it's likely I will release some alpha version next year.
A student government president campaign section for getting voters, features some video and my famous evil laugh. Muhahahahahahahahahahaha.
Inevitably, I will get a iRex iLiad for my birthday and finally save my eyes from reading long text from the LCD screen [backlight radiation hurts].

Jelly bean counting contest, or more precisely, guessing contest, can be seen at lot of places. My school's library start it annually. My third try is coming soon and I have gathered loads of experience, in fact, the 2nd one I joined, I was off only by 2 jelly beans, and the winner is also off by 2, but his number is lesser so he wins.

My goal is NOT to actually win the jelly beans, but HAVE the jelly beans.
What's the difference? Aren't the only way to get them is to win?
Not really... you will see it later.. my post name is misleading.. xD

There will be 3 level of seriousness in jelly bean counting contest, and I will explain each one of them in depth.

Level 1. idk, just lucky.
Nothing much to say about this level. Guess.

Level 2. I like to work 20 minutes on this...I didn't go to high school for nothing.
Let's assume you can only see the jar and the jar's content but can't touch it.
In this level, you want to approximate the number of jelly beans. Usual strategies:

1. Savvy~
If the jelly bean is in its original package, go to the super market and find one from the exact same brand, check out the net weight. and buy another jar of jelly bean from the same brand (smaller jar...), and weight individual jelly bean then find the average. divide the net weight by the average, you get approximately the same amount. Or you can get to the brand's website and it might just tell you how many jelly beans in that jar. Find out the previous year's winner's number, most of the times, people choses to use the same jelly bean brand.

2. Mathematica!
say, if the jelly beans are in a jar without any indication of it's brand. then you might have to use the volume method. This requires you to find the volume of the jar and the volume of individual jelly bean. To do that without measuring the jar directly(because you can't touches the jar, duh), try use trigonometry. Set up triangles by cast light from one point to the jar and then shadows on the wall, which then will be used to calculate the dimension of the jar by using similar triangles. same way to get the size of a jelly bean and then you are going to do the math. We know jelly beans can't tightly packed because it's not cubic shaped what should you do now? because we got the dimension, we can create a computer simulation program that randomly putting jelly beans into the jar and see how many come out each time1. I suggest the use of Mathematica. , take the number the arithmetic and geometric mean converges to. Basically, that will get a pretty good idea of how many jelly beans there are in the jar.

3. Classical Physics FTW^^
Ok, this is only useful if you can physically touch the jar...
Find out the force is acting on the jar. Think about the gravity, electromagnetic forces between atoms(here, both static friction and kinetic friction should be considered).
Now add a certain amount of force to the jar at a certain point for easy calculation. For example, project an simple shaped object with known mass and known velocity, find out the air friction coefficient to find the impact energy and then subtract the energy used for deformation. Or just push the jar with a weighing scale, find the distance the jar slide, and use the result found earlier, the kinetic friction(if no slide occurred, use static friction and find the lower bound of the jar's mass), calculate for the mass of the jar.
Now, to find out thee individual jelly bean's mass, you hav to consider momentum, record the movement of a jelly bean contrast with the movement of the jar and... figure the rest out.
It's also possible to use other methods, like using ideas from fluid dynamics, as long as it's possible to find out the mass of the jar.

Level 3. I'm deadly serious: power of sociality
It's Web 2.0 age now... the world is about social...

1. The power of other Level 1 contestants
I mean, even with everything I just told you, it's just not easy to get the exact number, and people might do the same thing and have better approximations. I wish you can have some more chances..
That's when people (usually, human) comes in
First, you have to do at least something in Level 2 so you can bribe the Level 1 contestants with "higher chance of winning but a bit lesser rewarding"2. show them the pyramid of risk and returns and convince them that it's worth it to stay in low risk. Then negotiate on how much jelly beans you will get if they win. After they agreed, you will become their representative and you assign them numbers. These playing cards allow a few tricks will be played out when you found out there is danger. The ones are not used to play tricks, spread them out in a small interval around the approximated number-- every 5 jelly bean one entry. And put your name down for the entry you think it's most likely to win.

2. Intelligence meets doom
Information is the most powerful weapon.
The most powerful act in the contest, is called the Sandwich of Actual Doom (SAD), relay on other player's count. Joe, choses 1500 to be his jelly bean count. It's extremely close to your approximation. Sacrifice 2 contestant, you can ensure Joe have a minimal chance to win, sandwich Joe's count by adding 1501 and 1499 in the pile. Muhahahahahahaa...
Now, some people who is also in level 3, will create false information to create wastes in your playing cards, be sure to check for reliable sources, and use lie detection methods.
At the same time, you have to create a cloud of lies... be sure to learn how to anti-lie detection.

3. Internet is a great place...
Web 2.0 everything... create a website about the jelly bean contest and get enough people to join it(and make sure your opponent can't find it out). Post all the information you gathered on that site, and use a digg like method to see how popular each count number it is... and use it as a reference to the number you will chose. Blog about this, many people might want to join the fight and post very intelligent comments, like "replace the jar of the jelly beans when no one is looking... go home.. count it up... and put it back...". Facebook it, add it to events, notes, groups...

Final Suggestion:
Don't over do it...

Bonus:
Few tips to anti-jelly bean approximation:
1. irregular jar shape
2. does not fill the jar
3. different mass for individual jelly beans
4. different volume for individual jelly beans
5. different density for individual jelly beans(kinda like the combine of the above 2)
6. hide something inside the jelly beans so people can't see
7. not allowing any information other than the view of the jar be accessed. (video camera record the jar in all perspectives and put the video up instead of the jar)

1. 1. Generalization: n-dimension hyper-jar can fit how many n-dimension hyper-jelly beans if the position of beans are random
2. 2. Don't say "lower risk but lower returns", because it doesn't sound great

Remember the PHP contest I was talking about last post?
I entered and I believe I can get at least a 3rd prize, a pen.
The code will be private until the day contest ends, but you can see my algorithm now.

At first, I thought this can be converted into a geometric question, then I just have to find the optimal solution for a geometric problem. Oh, how I wish I had read the entire book about computational geometry.

I still manage to generalize it into some geometric problem.

Each word is a point in a plane. Each word is connected with another word when there is exactly one letter difference between those 2 words. My idea is start from the starting word and trace all the links until it find the ending word.

My first idea is to create a web, map out all the lines between the points, and then, find the shortest lengh.

Then, my idea change. I only calculate the links for current point and move to another point. Do that recursively. Any point that's already touched will not be touched again. Like the picture below, when the link touches the ending word, the algorithm stops.

The algorithm returns a valid result, but it is not always the shortest path.

Here is a example.
Suppose these words are valid
AAA AAB ABB AAC ABC CBC

we want to turn AAA into CBC
with my algorithm, it might end up like this
AAA->AAB->ABB->ABC->CBC
but the shortest path is
AAA->AAC->ABC->CBC
why would that happen? Because as soon as my algorithm find the path, it will end and not verify if is it the shortest. If I want to verify if it's the shortest, I will have to go check all other path until it reaches this depth.

That doesn't sound fast, so I discard my idea and turned to a new one.
A tree like algorithm. The only difference between this one and the other one, is this algorithm doesn't go recursively. The algorithm find all the point relate to the first one, then move on, and then fine ALL the point related to the points that related to the first point(confusing...) BEFORE it goes into the next level. Now, when a link connects a word with the last word, it finds the shortest path.

Tim from vSig.org gave me an idea. Maybe it's possible to turn words into numbers and there might be some property between those numbers can find the shortest path to the word. I'm going to think that though.

The program is still slow... mainly because of the contest entries will be judged by "Does not produce any errors/warnings/notices."

Nah, I shouldn't blame them...it's slow because my implementation of the algorithm purposely1 include a LOT rehashing of arrays. So, I can be No.2 and get the t-shirt. I already have a Zend Studio, what can I do with the 2nd one? Give to a complex dog and let the real part eat it? Then wait for it to digest in it's imaginary stomach? and while I laugh at it, some old stereotypical dōjō dude come up and talk to the dog.
"Why are you so crazy, You no hungry for Zend Studio! you hungry for hot pockets!"

1. 1. I lied