https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Maybach&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T08:39:51ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Spam&diff=35630AoPS Wiki:Spam2010-08-13T01:59:00Z<p>Maybach: /* Spam on the AoPS message boards */</p>
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<div>'''Spam''' is a message, post, or article that is, for the most part, irrelevant. Context is the best guide for relevance.<br />
<br />
<br />
==Spam on the AoPS message boards==<br />
Spam on the [http://www.artofproblemsolving.com/Forum/index.php?f=155 AoPS message boards] can usually be classified as one of two major types. The first is advertising and the second is to raise one's post count or annoying or unproductive. The former is grounds for an immediate ban from the message boards and will always be deleted when caught. The second form is generally looked down upon, but the consequences are not so harsh.<br />
<br />
==Examples of Spam==<br />
<br />
Spam can include:<br />
-Repeating the same word over and over<br />
Example: "Hi hi hi hi hi hi hi hi hi"<br />
-Stating something off topic<br />
"What is the binomial Theorem"<br />
Spammer: "3.14159 I like pie. :) "<br />
-Going against ToS guidelines<br />
"**** *****"<br />
-Advertising<br />
"Buy Blah Blah shoes!"<br />
<br />
==Spam on AoPSWiki==<br />
Spam on [[Main_Page|AoPSWiki]] is monitored much more closely than that on the AoPS message boards. Absolutely no spam is allowed! Examples of spam on AoPSWiki would be writing stuff that is irrelevant to a topic in that particular article, posting links to irrelevant websites, etc.<br />
<br />
However, the AoPSWiki is not opposed to relevant commercial links, so long as they are appropriately placed and topical.<br />
<br />
The following is always considered spam:<br />
* Links to websites that are collections of links and advertisements. Just link the most appropriate links where they belong in the [[Main_Page|AoPSWiki]].<br />
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[[Category:AoPSWiki|Spam]]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Algebra&diff=33151Algebra2010-01-05T01:57:15Z<p>Maybach: /* Recommended AoPS books */</p>
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<div>In [[mathematics]], '''algebra''' can denote many things. As a subject, it generally denotes the study of calculations on some set. In high school, this can the study of examining, manipulating, and solving [[equation]]s, [[inequality|inequalities]], and other [[mathematical expression]]s. Algebra revolves around the concept of the [[variable]], an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]].<br />
<br />
Modern algebra (or "higher", or "abstract" algebra) deals (in part) with generalisations of the normal operations seen arithmetic and high school algebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common objects of study in higher algebra.<br />
<br />
As if to add to the confusion, "[[algebra (structure) |algebra]]" is the name for a certain kind of structure in modern algebra.<br />
<br />
Modern algebra also arguably contains the field of [[number theory]], which has important applications in computer science. (It is commonly claimed that the NSA is the largest employer in the USA of mathematicians, due to the applications of number theory to cryptanalysis.) However, number theory concerns itself with a specific structure (the [[ring]] <math>\mathbb{Z}</math>), whereas algebra in general deals with general classes of structure. Furthermore, number theory interacts more specifically with<br />
certain areas of mathematics (e.g., [[analysis]]) than does algebra in general. Indeed, number theory<br />
is traditionally divided into different branches, the most prominent of which are<br />
[[algebraic number theory]] and [[analytic number theory]].<br />
<br />
== Study Guides to Algebra ==<br />
<br />
* [[Algebra/Introduction | Introductory topics in algebra]]<br />
* [[Algebra/Intermediate | Intermediate topics in algebra]]<br />
* [[Algebra/Olympiad | Olympiad topics in algebra]]<br />
* [[Algebra/Advanced topics | More advanced topics in algebra]]<br />
<br />
== Recommended AoPS books ==<br />
*Introduction to Algebra [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=200]<br />
*Intermediate Algebra [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=300]<br />
Competition math for middle school<br />
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== See also ==<br />
* [[Abstract algebra]]<br />
* [[Elementary algebra]]<br />
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{{stub}}<br />
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[[Category:Algebra]] [[Category:Mathematics]]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=User_talk:Maybach&diff=33115User talk:Maybach2009-12-30T02:15:15Z<p>Maybach: /* "Area of a rhombus" */</p>
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<div>I am maybach and am having fun. :P</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=User_talk:Maybach&diff=33114User talk:Maybach2009-12-30T02:15:06Z<p>Maybach: /* "Area of a rhombus" */</p>
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<div>I am maybach and am having fun. :P<br />
<br />
== "Area of a rhombus" ==<br />
"I think you suck" jjx1</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Mathematics_websites&diff=33097Mathematics websites2009-12-27T01:32:14Z<p>Maybach: /* Websites for math enthusiasts */</p>
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<div>There are many great '''Mathematics websites''' around the [[internet]]. Here we organize a list of those sites we feel are best for students with high interest in [[mathematics]]. Note that [[mathematics forums]] are listed and discussed seperately.<br />
<br />
<br />
<br />
== Internet Resource Websites ==<br />
<br />
* [http://archives.math.utk.edu/index.html Math Archives]<br />
* [http://www.math-atlas.org/ Math Atlas]<br />
<br />
<br />
== Websites for high school students ==<br />
<br />
* [http://www.pims.math.ca/pi/ Pi in the Sky] is a mathematics magazine for high school students.<br />
* [http://mathideas.org/ Mathematical Ideas] is a website run by [[User:Chess64|Adeel Khan]].<br />
<br />
=== Websites for Olympiad students ===<br />
<br />
* [[Komal]] is a storied Hungarian [[math]] and [[physics]] journal. [http://www.komal.hu/info/bemutatkozas.e.shtml website].<br />
* [http://www.math.ust.hk/excalibur/ Mathematical Excalibur].<br />
* [http://reflections.awesomemath.org/ Mathematical Reflections] is a new online journal for Olympiad and collegiate mathematics.<br />
* [http://www.geometer.org/mathcircles/ Tom Davis's] site for [[math circles]] topics.<br />
* [http://mathworld.wolfram.com/ MathWorld] is a vast and well-maintained resource for math, science, and computer science professionals and students studying at a high level.<br />
<br />
== Websites for math enthusiasts ==<br />
<br />
* [[AoPS]] -- That's where you are now! [http://www.artofproblemsolving.com homepage].<br />
* [[Cut-the-knot]], a.k.a. Interactive Mathematics Miscellany and Puzzles, is a large and amazing site put together by [[Alexander Bogomolny]]. It includes an enormous number of [[mathematics articles]] and [[math games]] that are well-designed for teaching mathematical concepts. [http://www.cut-the-knot.org/index.shtml website].<br />
* [[Mathematical Database]] contains lots of useful math stuff. [http://eng.mathdb.org/ website].<br />
* [http://www.geometer.org/mathcircles/ Tom Davis's] site for [[math circles]] topics.<br />
* [ http://www.mathisfun.com] site for math<br />
<br />
== Websites for Math Teachers ==<br />
* [http://www.ct4me.net/ CT4ME] is dedicated to promoting the use of [[technology]] in [[mathematics education]].<br />
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<br />
== Websites for math history ==<br />
<br />
* [http://turnbull.dcs.st-and.ac.uk/history/index.html MacTudor History of Mathematics]<br />
* [[Wikipedia]] includes an enormous amount of information on the [http://en.wikipedia.org/wiki/History_of_math history of mathematics].<br />
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<br />
== Websites for mathematicians ==<br />
<br />
* [http://mathworld.wolfram.com/ MathWorld] is a vast and well-maintained resource for math, science, and computer science professionals and students studying at a high level.<br />
* [[Wikipedia]] includes articles about noncontroversial, published [http://en.wikipedia.org/wiki/Mathematics mathematics].<br />
* [http://www.math-atlas.org/ Math Atlas]<br />
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== See also ==<br />
<br />
* [[Mathematics competitions]]<br />
* [[Mathematics forums]]<br />
* [[Mathematics news]]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Writing_scholarships&diff=33036Writing scholarships2009-12-14T22:35:21Z<p>Maybach: /* Graduate */</p>
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<div>The following list of '''writing scholarships''' is primarily for American [[writing]] students. This list can be reorganized to incorporate scholarship programs in other countries. Just make that reorganization as clear and as clean as possible.<br />
<br />
Additions to this list are welcomed and encouraged. Please don't be stingy about letting students in on how they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[user:MCrawford | Mathew Crawford]] using the email crawford@artofproblemsolving.com with the scholarship listing you wish to contribute.<br />
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== Scholarship Search Sites ==<br />
*[http://apps.collegeboard.com/cbsearch_ss/welcome.jsp College Board Scholarship Search]<br />
*[http://www.fastweb.com FastWeb Scholarship search]<br />
<br />
== General Writing Scholarships ==<br />
*ALBA [http://www.alba-valb.org/albaeduess.htm Writing Scholarship]<br />
*Bridgestone Safety Scholarhip [http://www.safetyscholars.com/]<br />
*Knight Ridder [http://www.kri.com/career/internships.html#minority Writing Scholarships]<br />
*OP Loftbed [http://www.oploftbed.com/scholarship/application2006.php Writing Scholarship]<br />
<br />
== National Writing Scholarships ==<br />
*[http://www.aynrand.org/site/PageServer?pagename=education_contests_index Ayn Rand Foundation Essay Contests]<br />
*[http://us.penguingroup.com/static/html/services-academic/essayhome.html Penguin Group Signet Classics Student Scholarship Essay Contest]<br />
*[http://www.newsweekeducation.com/myturn2006/index.php Newsweek My Turn Essay Contest]<br />
*[http://www.scholastic.com/artandwritingawards/index.htm Scholastic Inc. Art & Writing Awards]<br />
<br />
== Scholarships by University ==<br />
*Appalachian State University [http://www.english.appstate.edu/scholarship.html Writing Scholarships]<br />
*Colorado State University [http://sfs.colostate.edu/J26020.cfm Writing Scholarship]<br />
*Mississippi State University [http://www.msstate.edu/dept/english/Schlshp.html Writing Scholarships]<br />
*Missouri State University [http://www.missouristate.edu/english/scholarships.htm Writing Scholarships]<br />
*Pittsburg State University [http://www.pittstate.edu/engl/scholarships.html Writing Scholarships]<br />
<br />
<br />
== Graduate ==<br />
www.graduatestudent.com</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Chicken_McNugget_Theorem&diff=32799Chicken McNugget Theorem2009-09-12T01:00:24Z<p>Maybach: /* Introductory */</p>
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<div>The '''Chicken McNugget Theorem''' states that for any two [[relatively prime]] [[positive integer]]s <math>m,n</math>, the greatest integer that cannot be written in the form <math>am + bn</math> for [[nonnegative]] integers <math>a, b</math> is <math>mn-m-n</math>.<br />
<br />
==Proof==<br />
Consider the integers <math>\pmod{m}</math>. Let <math>R = \{0, n, 2n, 3n, 4n ... (m-1)n\}</math>. Note that since <math>m</math> and <math>n</math> are relatively prime, <math>R</math> is a [[Complete residue system]] in modulo <math>m</math>.<br />
<br />
Lemma:<br />
For any given residue class <math>S \pmod{m}</math>, call <math>r</math> the member of <math>R</math> in this class. All members greater than or equal to <math>r</math> can be written in the form <math>am+bn</math> while all members less than <math>r</math> cannot for nonnegative <math>a,b</math>.<br />
<br />
Proof:<br />
Each member of the residue class can be written as <br />
<math>am + r</math> for an integer <math>a</math>. Since <math>r</math> is in the form <math>bn</math>, this can be rewritten as <math>am + bn</math>.<br />
Nonnegative values of <math>a</math> correspond to members greater than or equal to <math>r</math>. Negative values of <math>a</math> correspond to members less than <math>r</math>. Thus the lemma is proven.<br />
<br />
The largest member of <math>R</math> is <math>(m-1)n</math>, so the largest unattainable score <math>p</math> is in the same residue class as <math>(m-1)n</math>.<br />
<br />
The largest member of this residue class less than <math>(m-1)n</math> is <math>(m-1)n - m = mn - m - n</math> and the proof is complete.<br />
<br />
==Problems==<br />
===Introductory===<br />
Marcy buys paint jars in containers of 2 and 7. What's the largest number of paint jars that Marcy can't obtain?<br />
<br />
===Intermediate===<br />
Ninety-four bricks, each measuring <math>4''\times10''\times19'',</math> are to stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contribues <math>4''\,</math> or <math>10''\,</math> or <math>19''\,</math> to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks? [[1994 AIME Problems/Problem 11|Source]]<br />
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===Olympiad===<br />
<br />
<br />
==See Also==<br />
*[[Theorem]]<br />
*[[Prime]]<br />
<br />
{{stub}}<br />
[[Category:Theorems]]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=32786MATHCOUNTS2009-09-07T23:40:54Z<p>Maybach: /* State Competition */</p>
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<div>'''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [[CNA Foundation]], [[National Society of Professional Engineers]], the [[National Council of Teachers of Mathematics]], and others, the focus of MATHCOUNTS is on [[mathematical problem solving]]. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
Trigonometry, calculus and complex-valued functions may be utilized to achieve solutions although no problem would explicitly pertain to these notions, generally considered too advanced for middle schoolers.<br />
<br />
==Past Winners==<br />
* 1984: Michael Edwards, Texas<br />
* 1985: Timothy Kokesh, Oklahoma<br />
* 1986: Brian David Ewald, Florida<br />
* 1987: Russell Mann, Tennessee<br />
* 1988: Andrew Schultz, Illinois<br />
* 1989: Albert Kurz, Pennsylvania<br />
* 1990: Brian Jenkins, Arkansas<br />
* 1991: Jonathan L. Weinstein, Massachusetts<br />
* 1992: Andrei C. Gnepp, Ohio<br />
* 1993: Carleton Bosley, Kansas<br />
* 1994: William O. Engel, Illinois<br />
* 1995: Richard Reifsnyder, Kentucky<br />
* 1996: Alexander Schwartz, Pennsylvania<br />
* 1997: Zhihao Liu, Wisconsin<br />
* 1998: Ricky Liu, Massachusetts<br />
* 1999: Po-Ru Loh, Wisconsin<br />
* 2000: Ruozhou Jia, Illinois<br />
* 2001: Ryan Ko, New Jersey<br />
* 2002: Albert Ni, Illinois<br />
* 2003: Adam Hesterberg, Washington<br />
* 2004: Gregory Gauthier, Illinois<br />
* 2005: Neal Wu, Louisiana (Neal is a user on AoPS under the username [[User:nebula42|nebula42]])<br />
* 2006: Daesun Yim, New Jersey (Daesun is a user on AoPS under the usernames [[User:Treething|Treething]] and [[User:Lazarus|Lazarus]])<br />
* 2007: Kevin Chen, Texas (Kevin is a user on AoPS under the username [[User:binonunquineist|binonunquineist]])<br />
* 2008: Darryl Wu, Washington (youngest winner ever, at 11, as well as the first 6th grader to ever even make the National Countdown Round)<br />
* 2009: Bobby Shen, Texas (Bobby is a user on AoPS under the username [[User:stevenmeow|stevenmeow]])<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
<br />
=== Sprint Round ===<br />
<br />
30 problems in 40 minutes. This round is generally made up questions ranging from relatively easy to relatively difficult. Some of the difficult problems are only difficult because calculators are not allowed in this round.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Each set of two problems is given six minutes. Students may not go back to previous rounds even if they finish before time is called. Students may use calculators.<br />
<br />
=== Team Round ===<br />
<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion.<br />
<br />
<br />
=== Ciphering Round ===<br />
In some states, (most notably Florida) there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
====Chapter and State Competitions====<br />
<br />
In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
<br />
*The winner of the first round goes up against the 8th place finisher.<br />
<br />
*The winner of the second round goes up against the 7th place finisher.<br />
<br />
This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three question correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field.<br />
<br />
At the first round and the second round, the first person to correctly answer three questions wins. However, at the semifinals, the rules slightly change&mdash;the first person to correctly answer four questions wins.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to challenging problems. This round is optional at the state level competition and is mandatory at the national competition. At nationals the top two on the written and countdown participate.<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of 30 + 2(8) = 46 points.<br />
<br />
A team's score is the average of the individual scores of its four members plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score.<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level require the least depth of curriculum.<br />
<br />
=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size. In California, this is the most difficult round. Questions are relatively easy compared to the National Competition.<br />
<br />
=== State Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. At this level, the questions are getting significantly harder, and it is rare for a lot of kids to score above 30/46.<br />
<br />
=== National Competition ===<br />
==== Nation Competition Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2009 competition was held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
<br />
==== Rewards ====<br />
<br />
Every competitor at the national competition receives a graphing calculator that varies by year - for example, in 2006 it was a TI-84 Plus Silver Edition with the MATHCOUNTS logo on the back. In 2007, MATHCOUNTS took the logo off. In 2008 and 2009, they gave TI-<math>n</math>spires to everyone. They also give out a laptop and an 8000 dollar scholarship to the winner.<br />
<br />
== MATHCOUNTS Resources ==<br />
=== MATHCOUNTS Books ===<br />
* [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php MATHCOUNTS books] at the [http://www.artofproblemsolving.com/Books/AoPS_B_About.php AoPS Bookstore]<br />
* [[Art of Problem Solving]]'s [http://www.artofproblemsolving.com/Books/AoPS_B_Rec_Middle.php Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS.<br />
<br />
=== MATHCOUNTS Classes ===<br />
* [[Art of Problem Solving]] hosts [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesP.php#mc MATHCOUNTS preparation classes].<br />
* [[Art of Problem Solving]] hosts many free MATHCOUNTS [[Math Jams]]. [http://www.artofproblemsolving.com/Community/AoPS_Y_Math_Jams.php Math Jam Schedule]. [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php Math Jam Transcript Archive].<br />
<br />
=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org MATHCOUNTS Homepage]<br />
* [[Art of Problem Solving]] hosts a large [http://www.artofproblemsolving.com/Forum/index.php?f=132 MATHCOUNTS Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, 2008 and 2009 National Champion [[Texas MathCounts]] teams.<br />
<br />
== What comes after MATHCOUNTS? ==<br />
<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
<br />
[[Category:Mathematics competitions]]<br />
<br />
== See also ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[MATHCOUNTS historical results]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Rocket_City_Math_League&diff=32785Rocket City Math League2009-09-07T23:38:08Z<p>Maybach: /* See also */</p>
<hr />
<div>The '''Rocket City Math League''' is a national [[mathematics competition]] for middle and high school students run by students at [[Virgil I. Grissom High School]] in Huntsville, Alabama. [http://www.rocketcitymath.org/index2.htm Homepage]<br />
<br />
<br />
== See also ==<br />
* [http://www.artofproblemsolving.com/Forum/index.php?f=305 RCML Forum] at [[Art of Problem Solving]]<br />
* [[Mathematics competition resources]]<br />
<br />
*The contest is divided into several categories: Pre Algebra(Explorer), Algebra I(Mercury), Geometry(Gemini), Algebra II(Apollo), and Comprehensive(Discovery).<br />
Problems range from calculus in Discovery to topics covered in 5th or 6th grade in Explorer<br />
{{stub}}</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Proof_writing&diff=32784Proof writing2009-09-07T23:33:46Z<p>Maybach: /* Practice */</p>
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<div>'''Proof writing''' is often thought of as one of the most difficult aspects of math education to conquer. Proofs require the ability to think abstractly, that is, universally. They also require a little appreciation for mathematical culture; for instance, when a mathematician uses the word "trivial" in a proof, they intend a different meaning to how the word is understood by the wider population. Students who spend time studying maths can develop proof-writing skills over time.<br />
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== Getting Started ==<br />
The fundamental aspects of a good proof are precision, accuracy, and clarity. A single word can change the intended meaning of a proof, so it is best to be as precise as possible. <br />
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There are two different types of proofs: informal and formal.<br />
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[[Formal proof]] is often introduced using a [[two-column proof|two-column format]], as favored by many geometry teachers. In higher-level mathematics (taken as meaning an advanced undergraduate level of mathematical maturity or above), two methods of formal proof predominate. These are [[proof by construction]] (a common example of which is [[induction|proof by induction]]), and [[proof by contradiction]] (which in its simplest form requires only the demonstration of a [[counterexample]]).<br />
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An [[informal proof]] can be in a wide variety of styles. It is usually not as neat as a two-column proof but is far easier to organize. It is important to note that people trained to a university-level of mathematics do not consider so-called "informal proofs" to be proof of anything at all. Instead, they may be regarded as "heuristics", or teaching tools, at best.<br />
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=== Practice ===<br />
[[Art of Problem Solving]] (AoPS) has many resources to help students begin writing proofs. <br />
* The AoPS forums (which you can get to through the '''Community''' tab on the left sidebar) are a great place to practice writing solutions to problems. Do your best to make your explanations both clear and complete. Read solutions by other students to see what you might do better. Listen to the constructive criticisms of others.<br />
* AoPS Blogs (also in the Community area) are a great place to showcase your best solutions.<br />
* The [[AoPSWiki]] you are in now is written by members of the AoPS community. Contributing to the AoPSWiki means writing mathematics as clearly as you can.<br />
* The Contests button on the top of the page has many Olympiad-level problems from contests such as IMO, USAMO, and many international mathematics competitions.<br />
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== Proof Writing Guides ==<br />
* [http://www.artofproblemsolving.com/Resources/AoPS_R_A_HowWrite.php How to Write a Solution] by [[Richard Rusczyk]] and [[Mathew Crawford]]<br />
* [http://www.stonehill.edu/compsci/History_Math/math-read.htm How to Read Mathematics] -- Not really proof writing, but a helpful read for those learning to write basic proofs.<br />
* [http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1218677186&sr=8-1 How To Prove It: A Structured Approach] by Daniel J. Velleman -- an excellent primer on methods of proof; train your ability to do proofs by induction, contradiction and more.<br />
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== See Also ==<br />
* [[Math books]]<br />
* [[Mathematics competitions]]<br />
* [[Mathematics competition resources]]<br />
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[[Category:Proofs]] [[Category:Mathematics]]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Roman_numerals&diff=32781Roman numerals2009-09-07T16:37:16Z<p>Maybach: /* History */</p>
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<div>{{stub}}<br />
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1 - '''I''' ''(unus)''<br />
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5 - '''V''' ''(quinque)''<br />
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10 - '''X''' ''(decem)''<br />
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50 - '''L'''<br />
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100 - '''C''' ''(centum)''<br />
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500 - '''D'''<br />
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1000 - '''M''' ''(mille)''<br />
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Numbers are written as combinations of letters. Letters are written from biggest to smallest value. Instead of writing IIII for 4, IV is used. To "subtract," a smaller-value letter is placed before a larger-value letter. This done to make numbers smaller (e.g. IX instead of VIIII).<br />
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==History==<br />
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The ancient Romans developed a numeration system 3000 years ago. This system, now known as Roman numerals, lasted several centuries as the main numeration system. Today, Roman numerals are still used, but not as much as they were 3000 years ago.<br />
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You can also add a line on top to make it be multiplied by 1,000.<br />
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==Examples==<br />
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513 - DXIII<br />
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99 - IC<br />
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2006 - MMVI<br />
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==See Also==<br />
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* [[Number systems]]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Base_numbers&diff=32780Base numbers2009-09-07T16:36:27Z<p>Maybach: /* Beginner */</p>
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<div>To understand the notion of '''base numbers''', we look at our own [[number system]]. We use the [[decimal]], or base-10, number system. To help explain what this means, consider the number 2746. This number can be rewritten as <math>\displaystyle 2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math><br />
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Note that each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. The first digit to the left of the decimal place (recall that the decimal place is to the right of the 6, i.e. 2746.0) tells us that there are six <math>10^0</math>'s, the second digit tells us there are four <math>10^1</math>'s, the third digit tells us there are seven <math>10^2</math>'s, and the fourth digit tells us there are two <math>\displaystyle 10^3</math>'s.<br />
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Base-10 uses digits 0-9. Usually, the base, or '''radix''', of a number is denoted as a subscript written at the right end of the number (e.g. in our example above, <math>2746_{10}</math>, 10 is the radix).<br />
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== Base Number Topics ==<br />
* [[base numbers/Common bases | Common bases]]<br />
* [[base numbers/Conversion | Converting between bases]]<br />
* [[Improper fractional base]]<br />
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== History ==<br />
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Base-10 is an apparently obvious counting system because people have 10 fingers. Historically, different societies utilized other systems. The Babylonian cultures are known to have used base-60; this is why we say there are 360 degrees in a circle and (fact check on this one coming) why we count 60 minutes in an hour and 60 seconds in a minute (they might have used it because it has so many multiples, 12 in fact, we wouldn't want any fractions). The [[Roman system]], which didn't have any base system at all, used certain letters to represent certain values (e.g. I=1, V=5, X=10, L=50, C=100, D=500, M=1000). Imagine how difficult it would be to multiply LXV by MDII! That's why the introduction of the '''Arabic numeral system''', base-10, revolutionized math and science in Europe.<br />
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== Example Problems ==<br />
=== Beginner ===<br />
*Evaluate <math>\sqrt{61_{8}}</math> as a number in the decimal system.<br />
**Solution: <math>61_{8}</math> must be rewritten in the decimal system (base-10) before evaluating the square root. To do this, multiply and add <math>6*8^1+1*8^0=48+1=49. \sqrt{49}=7.</math> Therefore, the answer is 7.<br />
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Find the base 2 numner that is equivalent to 49(base 7)<br />
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=== Intermediate ===<br />
* [[2003_AIME_I_Problems/Problem_13 | 2003 AIME I Problem 13]]<br />
* [[1977_Canadian_MO_Problems/Problem_3 | Canadian Mathematics Olympiad Problem 3]]<br />
* Suppose <math>P(x)</math> is an unknown polynomial, of unknown degree, with nonnegative integer coefficients. Your goal is to determine this polynomial. You have access to an oracle that, given an integer <math>n</math>, spits out <math>P(n)</math>, the value of the polynomial at <math>n</math>. However, the oracle charges a fee for each such computation, so you want to minimize the number of computations you ask the oracle to do. Show that it is possible to uniquely determine the polynomial after only two consultations of the oracle. ([http://www.math.uiuc.edu/~hildebr/pow/pow10.pdf UIUC POW])<br />
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== Resources ==<br />
==== Books ====<br />
* The AoPS [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=10 Introduction to Number Theory] by [[Mathew Crawford]].<br />
==== Classes ====<br />
* [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php#begnum AoPS Introduction to Number Theory Course]<br />
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== See Also ==<br />
*[[Number theory]]<br />
*[[Modular arithmetic]]<br />
*[http://www.artofproblemsolving.com/Forum/weblog_entry.php?t=92951 Richard Rusczyk's Base Number Article]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=Pigeonhole_Principle&diff=32779Pigeonhole Principle2009-09-07T16:31:21Z<p>Maybach: /* Introductory Problems */</p>
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<div>Also known as the [[Dirichlet]] box principle, <br />
the basic '''pigeonhole principle''' states that if there are <math>n</math> boxes, and more than <math>{n}</math> objects, then one box must contain two or more objects. <br />
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The extended version of the pigeonhole principle states that for <math>n</math> boxes, and more than <math>{nk}</math> objects, some box must contain at least <math>k+1</math> objects. Although this theorem seems obvious, many challenging olympiad problems can be solved by applying the pigeonhole principle.<br />
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== Examples ==<br />
=== Introductory Problems ===<br />
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If a Martian has an infinite number of red, blue, yellow, and black socks, how many socks must the martian pull out to guarantee he has a pair?<br />
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=== Intermediate Problems ===<br />
# Show that in any group of five people, there are two who have an identical number of friends within the group. ([[Pigeonhole Principle/Solutions#M1|Solution]]) <div style="text-align:right">(Mathematical Circles)</div> <br />
# Six distinct positive integers are randomly chosen between 1 and 2006, inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5? ([[Pigeonhole Principle/Solutions#M2|Solution]])<br />
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<math>\mathrm{(A) \ } \frac{1}{2}\qquad\mathrm{(B) \ } \frac{3}{5}\qquad\mathrm{(C) \ } \frac{2}{3}\qquad\mathrm{(D) \ } \frac{4}{5}\qquad\mathrm{(E) \ } 1\qquad</math><br />
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<div style="text-align:right">([[2006 AMC 10A Problems/Problem 20]])</div><br />
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=== Olympiad Problems ===<br />
# Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle. ([[Pigeonhole Principle/Solutions#O1|Solution]]) <div style="text-align:right">(Manhattan Mathematical Olympiad 2004)</div><br />
# Prove that having 100 whole numbers, one can choose 15 of them so that the difference of any two is divisible by 7. ([[Pigeonhole Principle/Solutions#O2|Solution]]) <div style="text-align:right">(Manhattan Mathematical Olympiad 2005)</div><br />
# Prove that from any set of one hundred whole numbers, one can choose either one number which is divisible by 100, or several numbers whose sum is divisible by 100. ([[Pigeonhole Principle/Solutions#O3|Solution]]) <div style="text-align:right">(Manhattan Mathematical Olympiad 2003)</div><br />
# Prove that among any ten points located on a circle with diameter 5, there exist at least two at a distance less than 2 from each other. ([[Pigeonhole Principle/Solutions#O4|Solution]]) <div style="text-align:right">(Japan 1997)</div><br />
#Every point in a plane is either red, green, or blue. Prove that there exists a rectangle in the plane such that all of its vertices are the same color. ([[Pigeonhole Principle/Solutions#O5|Solution]]) <div style="text-align:right">(USAMTS Year 18 - Round 1 - Problem 4)</div><br />
# There are 51 senators in a senate. The senate needs to be divided into <math>n</math> committees such that each senator is on exactly one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does 'not' necessarily hate senator A.) Find the smallest <math>n</math> such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee. ([[Pigeonhole Principle/Solutions#O6|Solution]]) <div style="text-align:right">(Red [[MOP]] lecture 2006)</div><br />
# Show that for any <math>{x\in\mathbb R}</math> and positive integer <math>{n}</math>, there exists a rational number <math>{\frac pq}</math> with <math>1\le q\le n</math> such that <math>\left|x-\frac pq\right|<\frac 1{nq}.</math> ([[Pigeonhole Principle/Solutions#O7|Solution]]) <div style="text-align:right">(the classical Rational Approximation Theorem)</div><br />
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== See also ==<br />
* [[Combinatorics]]<br />
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[[Category:Combinatorics]]</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=User_talk:Maybach&diff=32694User talk:Maybach2009-08-19T19:05:31Z<p>Maybach: </p>
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<div>I am maybach and am having fun. :P</div>Maybachhttps://artofproblemsolving.com/wiki/index.php?title=User_talk:Maybach&diff=32578User talk:Maybach2009-08-07T22:38:08Z<p>Maybach: ugh</p>
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<div>hi!</div>Maybach