https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=MysticTerminator&feedformat=atomAoPS Wiki - User contributions [en]2021-08-03T22:46:43ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=Algebraic_geometry&diff=27704Algebraic geometry2008-09-02T03:27:57Z<p>MysticTerminator: </p>
<hr />
<div>'''Algebraic geometry''' is the study of solutions of [[polynomial]] equations by means of [[abstract algebra]], and in particular [[ring theory]]. Algebraic geometry is most easily done over [[algebraically closed]] [[field]]s, but it can also be done more generally over any field or even over [[ring]]s. It is not to be confused with [[analytic geometry]], which is use of coordinates to solve geometrical problems.<br />
<br />
== Affine Algebraic Varieties ==<br />
<br />
One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^k</math> denote [[affine]] <math>k</math>-space, i.e. a [[vector space]] of [[dimension]] <math>k</math> over an algebraically closed field, such as the field <math>\mathbb{C}</math> of [[complex number]]s. (We can think of this as <math>k</math>-dimensional "complex Euclidean" space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_k]</math> be the [[polynomial ring]] in <math>k</math> variables, and let <math>I</math> be a [[maximal ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^k\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''.<br />
<br />
== Projective Varieties ==<br />
Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties.<br />
<br />
== Quasiprojective Varieties ==<br />
<br />
The varieties most commonly used, quasiprojective varieties are algebraic varieties given as open subsets of a projective variety with respect to the Zariski topology.<br />
<br />
== General Algebraic Varieties ==<br />
<br />
Defined in terms of sheafs and patchings. <br />
<br />
== Schemes ==<br />
Let <math>A</math> be a ring and <math>X=\operatorname{Spec}A</math>. An affine scheme is a ringed topological space isomorphic to some <math>(\operatorname{Spec }A,\mathcal{O}_{\operatorname{Spec}A})</math>.<br />
A scheme is a ringed topological space <math>(X,\mathcal{O}_X)</math> admitting an open covering <math>\{U_i\}_i</math> such that <math>(U_i,\mathcal{O}_{X|U_i})</math> is an affine scheme for every <math>i</math>.<br />
<br />
{{stub}}<br />
<br />
[[Category:Algebra]]<br />
[[Category:Geometry]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=AoPS_Community_Awards&diff=26849AoPS Community Awards2008-07-02T07:08:12Z<p>MysticTerminator: New page: what are AoPS community awards? it shows up on the banner at the top of the site from time to time ... shouldn't there be something here.</p>
<hr />
<div>what are AoPS community awards? it shows up on the banner at the top of the site from time to time ... shouldn't there be something here.</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Arithmetic_Mean-Geometric_Mean_Inequality&diff=26802Arithmetic Mean-Geometric Mean Inequality2008-06-27T22:45:28Z<p>MysticTerminator: /* Extensions */</p>
<hr />
<div>{{WotWAnnounce|week=June 27-July 3}}<br />
<br />
The '''Arithmetic Mean-Geometric Mean Inequality''' ('''AM-GM''' or '''AMGM''') is an elementary [[inequality]], and is generally one of the first ones taught in inequality courses.<br />
<br />
== Theorem ==<br />
AM-GM states that for any [[set]] of [[nonnegative]] [[real number]]s, the [[arithmetic mean]] of the set is greater than or [[equal]] to the [[geometric mean]] of the set. Algebraically, this is expressed as follows.<br />
<br />
For a set of nonnegative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following always holds:<br />
<cmath> \frac{a_1+a_2+\ldots+a_n}{n}\geq\sqrt[n]{a_1a_2\cdots a_n} </cmath><br />
Using the shorthand notation for [[summation]]s and [[product]]s:<br />
<cmath> \sum_{i=1}^{n}a_i}/n \geq \prod\limits_{i=1}^{n}a_i^{1/n} . </cmath><br />
For example, for the set <math>\{9,12,54\}</math>, the arithmetic mean, 25, is greater than the geometric mean, 18; AM-GM guarantees this is always the case. <br />
<br />
The [[equality condition]] of this [[inequality]] states that the arithmetic mean and geometric mean are equal [[iff|if and only if]] all members of the set are equal.<br />
<br />
AM-GM can be used fairly frequently to solve [[Olympiad]]-level inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]].<br />
<br />
=== Proof ===<br />
<br />
There are so many proofs of AM-GM that they have an article to themselves: [[Proofs of AM-GM]].<br />
<br />
=== Weighted Form ===<br />
<br />
The weighted form of AM-GM is given by using [[weighted average]]s. For example, the weighted arithmetic mean of <math>x</math> and <math>y</math> with <math>3:1</math> is <math>\frac{3x+1y}{3+1}</math> and the geometric is <math>\sqrt[3+1]{x^3y}</math>.<br />
<br />
AM-GM applies to weighted averages. Specifically, the '''weighted AM-GM Inequality''' states that if <math>a_1, a_2, \dotsc, a_n</math> are nonnegative real numbers, and <math>\lambda_1, \lambda_2, \dotsc, \lambda_n</math> are nonnegative real numbers (the "weights") which sum to 1, then<br />
<cmath> \lambda_1 a_1 + \lambda_2 a_2 + \dotsb + \lambda_n a_n \ge a_1^{\lambda_1} a_2^{\lambda_2} \dotsm a_n^{\lambda_n}, </cmath><br />
or, in more compact notation,<br />
<cmath> \sum_{i=1}^n \lambda_i a_i \ge \prod_{i=1}^n a_i^{\lambda_i} . </cmath><br />
Equality holds if and only if <math>a_i = a_j</math> for all integers <math>i, j</math> such that <math>\lambda_i \neq 0</math> and <math>\lambda_j \neq 0</math>.<br />
We obtain the unweighted form of AM-GM by setting <math>\lambda_1 = \lambda_2 = \dotsb = \lambda_n = 1/n</math>.<br />
<br />
==Extensions==<br />
<br />
* The [[power mean inequality]] is a generalization of AM-GM which places the arithemetic and geometric means on a continuum of different means.<br />
* The [[root-square-mean arithmetic-mean geometric-mean harmonic-mean inequality]] is special case of the power mean inequality.<br />
* Kedlaya also extended it greatly doing some stuff like the arithmetic mean of the sequence of geometric means is at least the geometric mean of the sequence of arithmetic means or something like that.<br />
<br />
==Problems==<br />
<br />
=== Introductory ===<br />
=== Intermediate ===<br />
* Find the minimum value of <math>\frac{9x^2\sin^2 x + 4}{x\sin x}</math> for <math>0 < x < \pi</math>.<br />
([[1983 AIME Problems/Problem 9|Source]])<br />
=== Olympiad ===<br />
<br />
* Let <math>a </math>, <math>b </math>, and <math>c </math> be positive real numbers. Prove that<br />
<cmath> (a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \ge (a+b+c)^3 . </cmath><br />
([[2004 USAMO Problems/Problem 5|Source]])<br />
<br />
== See Also ==<br />
<br />
* [[RMS-AM-GM-HM]]<br />
* [[Algebra]]<br />
* [[Inequalities]]<br />
<br />
==External Links==<br />
* [http://www.mathideas.org/problems/2006/5/29.pdf Basic Inequalities by Adeel Khan]<br />
* [http://www.mathideas.org/problems/2006/5/31.pdf Inequalities: An Application of RMS-AM-GM-HM by Adeel Khan]<br />
<br />
<br />
[[Category:Inequality]]<br />
[[Category:Theorems]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Schrodinger_Equation&diff=26727Schrodinger Equation2008-06-20T20:57:59Z<p>MysticTerminator: </p>
<hr />
<div>In the Schrodinger picture, the equation governing quantum mechanical evolution of some state <math>\Psi</math> in the relevant Hilbert space is given by <math>i\hbar\partial_t\Psi = \hat{H}\Psi</math>, where <math>\hat{H}</math> is the linear operator representing the Hamiltonian, usually of the form <math>-\frac{\hbar^2}{2m}\Delta + V</math> where <math>\Delta</math> is the relevant Laplace(-Beltrami) operator.</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Schrodinger_Equation&diff=26726Schrodinger Equation2008-06-20T20:57:20Z<p>MysticTerminator: New page: In the Schrodinger picture, the equation governing quantum mechanical evolution of some state <math>\Psi</math> in the relevant Hilbert space is given by <math>i\hbar\partial_t\Psi = \hat{...</p>
<hr />
<div>In the Schrodinger picture, the equation governing quantum mechanical evolution of some state <math>\Psi</math> in the relevant Hilbert space is given by <math>i\hbar\partial_t\Psi = \hat{H}\Psi</math>, where <math>\hat{H}</math> is the linear operator representing the Hamiltonian.</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=William_Lowell_Putnam_Mathematical_Competition&diff=26724William Lowell Putnam Mathematical Competition2008-06-20T19:17:21Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=June 20-26}}<br />
<br />
The '''William Lowell Putnam Mathematical Competition''' is a highly challenging, proof-oriented [[mathematics competition]] for undergraduate students in North America.<br />
<br />
Top scoring students on the Putnam exam are named Putnam Fellows.<br />
<br />
== The Competition ==<br />
The '''Putnam Exam''' is a 6 hour undergraduate exam usually held the first Saturday in December. The test consists of two 3 hour sessions of six problems each with 2 hour lunch break between them. The problems are proof-oriented and written in roughly the same style as high school Olympiads are, although they include more advanced mathematics. Each problem is graded on a scale of 0 to 10, with nearly all scores falling in the ranges <math>[0, 3]</math> and <math>[8, 10]</math>. The top five scorers (or more if there are ties) on the exam are named "Putnam Fellows." <br />
<br />
Each school may have as many students as are interested sit for the exam. Before the contest, three students are selected as the official school Putnam team. Each team's score is determined by adding the ranks (not the scores) of the three students on the team, and the team with the lowest point total wins. For example, a school whose team members placed 1st, 2nd and 20th would place lower than a school whose team members placed 6th, 7th and 8th. (In the case of ties, every student is assigned the average of the range of ranks that would have been attained had there been no tie -- that is, if the top three students tie, they are all awarded a rank of <math>\frac{1 + 2 + 3}{3} = 2</math>.) The fact that the team members need to be chosen in advance regularly leads to schools selecting the "wrong team." For example, at least three of the six 2007 Putnam fellows were not members of their school team: AoPS member Arnav Tripathy of Harvard, Qingchun Ren of MIT, and Xuancheng Shao, also of MIT. In some cases this has led to a team placing lower than they would have had they chosen the three team members who went on to score the highest. In fact, this almost always has a noticeable effect on MIT's score, which leads to everyone else making fun of them for choosing horribly every single year.<br />
<br />
A person may take the Putnam Exam a maximum of four times. Typically, this means a student may sit each year he or she is an undergraduate, although high school seniors have occasionally taken the exam officially.<br />
<br />
==Placings and Prizes==<br />
The prizes are as follows:<br />
<br />
===Individuals===<br />
*Putnam fellows<br />
** <dollar/>2,500<br />
*The next top ten individuals<br />
** <dollar/>1,000<br />
*Next Ten Individuals <br />
** <dollar/>250<br />
<br />
===Teams===<br />
*First Place team<br />
**Team members recieve &#36;1,000<br />
**School recieves &#36;25,000<br />
*Second Place Team<br />
**Team members recieve 800<br />
**School recieves &#36;20,000<br />
*Third Place Team<br />
**Team members recieve &#36;600<br />
**School recieves &#36;15,000<br />
*Fourth Place Team <br />
**Team members recieves &#36;400 <br />
**School recieves &#36;10,000<br />
*Fifth Place Team <br />
**Team members recieve &#36;200<br />
**School recieves &#36;5,000<br />
<br />
*Elizabeth Lowell Putnam Prize- The Elizabeth Lowell Putnam Prize will be awarded periodically to a woman whose performance in the competition has been deemed particularly meritorious. This prize would be in addition to any other prize she might otherwise win. Women contestants, to be eligible for this prize, must specify their gender. <br />
**&#36;1,000<br />
<br />
<br />
<br />
== Problem Books ==<br />
* [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=47 1938-1964] -- A good book for students just learning to solve Putnam Problems.<br />
* [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=48 1965-1984]<br />
* [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?page_id=49 1985-2000] by [[Kiran Kedlaya]], [[Bjorn Poonen]], and [[Ravi Vakil]]. The three authors are among the most successful Putnam participants of all time.<br />
<br />
<br />
<br />
== Resources ==<br />
* [http://www.unl.edu/amc/a-activities/a7-problems/putnamindex.shtml Putnam Archive] on the [[AMC]] website.<br />
* <url>Forum/index.php?f=80 The Putnam Forum</url> at [[AoPS]].<br />
* <url>Forum/resources.php?c=2&cid=23 Putnam problem and solution database</url> in the <url>Forum/resources.php AoPS Contest Database</url>.<br />
* <url>Resources/AoPS_R_A_HowWrite.php How to Write a Solution</url> by [[Richard Rusczyk]] & [[user:MCrawford | Mathew Crawford]].<br />
* [[List of United States college mathematics competitions]]<br />
* [[Putnam historical results]]<br />
<br />
[[Category:Mathematics competitions]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Mathematical_Olympiad_Summer_Program&diff=26565Mathematical Olympiad Summer Program2008-06-16T00:14:43Z<p>MysticTerminator: /* History and Culture */</p>
<hr />
<div>{{WotWAnnounce|week=June 13-19}}<br />
<br />
The '''Mathematical Olympiad Summer Program''' (abbreviated '''MOP''') is a 3-week intensive problem solving camp held at the University of Nebraska-Lincoln to help high school students prepare for math olympaids, especially the [[International Mathematical Olympiad]]. While the program is free to participants, invitations are limited to the top finishers on [[USAMO]].<br />
<br />
== Purpose ==<br />
One purpose of MOP is to select and train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results of the USAMO and the TST are weighted equally when selecting the US IMO team.<br />
<br />
The other important purpose of MOP is to train younger students in Olympiad-level problem solving.<br />
<br />
== Information ==<br />
MOP is currently held at the University of Nebraska-Lincoln. While the dates vary from year to year, MOP is generally held in the last three weeks of June. This year the dates are June 10-July 3, 2008.<br />
<br />
Invitations are extended to the top non-Canadian finishers on USAMO. Students receiving invitations can be divided into four groups:<br />
<br />
USAMO winners: The Americans among the top 12 finishers on USAMO are invited to MOP regardless of their age. Additionally, they are invited to take the Team Selection Test and are viewed as potential members of the American IMO team for that year.<br />
<br />
Top non-senior USAMO finishers: In addition to the winners, the next 18 or so non-senior non-Canadian finishers are invited to attend MOP. This group is viewed as potential IMO team members for future years, although in extreme circumstances (including 2006) IMO team members for that year have been drawn from this pool.<br />
<br />
Top 30 freshman: The top 30 freshman on USAMO are invited to attend MOP with the goal of providing them with a foundation in olympiad level mathematics.<br />
<br />
For 2008, another group has been added. The girls who will be representing the United States at the Chinese Girls Math Olympiad will attend MOP to prepare for that contest.<br />
<br />
== Structure of the Program ==<br />
<br />
MOP is divided into three groups that roughly correspond with the first three kinds of invitations. Black MOP consists of that year's USAMO winners and contains the IMO team members and alternates. Blue MOP is for the second group of invitees and mostly consists of students who just completed their junior or sophomore year of high school, although in exceptional cases some 7th and 8th graders have participated. Finally, Red MOP consists of all the freshman who were invited to participate. Students and instructors have discretion in selecting which group they're part of and may choose to transfer part way through the program; this generally involves members of Black dropping down to Blue or occasionally members of Red promoting themselves to Blue. The three groups take classes and practice tests separately, are given different levels of material to practice with, and to a certain extent are distinct socially. For the most part, Blue and Black are more closely associated to each other than either one is to Red; for example, members of Blue and Black are placed on the same teams for the team contest.<br />
<br />
Each Weekday consists of three instructional sessions: 8:30 AM - 11:30 AM, 1:15 PM - 3:15 PM, and 8:00 PM - 10:00 PM. Classes usually consist of a lecture followed by a problem set. Solutions are often presented by students with the supervision of an instructor.<br />
<br />
Timed and graded olympiad style tests are an integral part of MOP. Every few days, a 4-hour, 3-question test is administered in place of the afternoon lecture, and is graded with comments within 2-3 days.<br />
<br />
Team tests also occur weekly. Students are divided into teams of five, and work on a set of twenty-five problems for approximately half a week. On the day of the contest, the teams present solutions to problems which have not yet been presented, in arbitrary order. The fun starts when all of the easy problems have been taken, and teams resort to certain creative methods in order to solve a problem.<br />
<br />
The combination of these makes MOP an extraordinarily intense experience. One participant at 2007 MOP calculated that by the end of the second week members of Blue MOP had already spent more time in a classroom than most calculus classes do in a year, and by the end of the third week participants had spent 170 hours over 19 days either in class or taking practice test for an average of roughly 9 hours a day of math- and that's before time spent doing problem sets and working on the team contest outside of class is included.<br />
<br />
== History and Culture ==<br />
MOP was created in 1974 as a training camp for the first United States IMO team. <br />
<br />
At the time that MOP was established the official name was simply "Mathematical Olympiad Program", which was the source of the original abbreviation "MOP". At some point, however, the official name was changed to "Mathematical Olympiad Summer Program" and the official abbreviation became "MOSP". Despite this change, participants and alumni almost universally continue to refer to the program as "MOP". Although some administrators continue to use "MOSP" in official documents, students use "MOP" in every setting. One former participant testifies, "Any lost souls using the other appellation are looked upon with pity and regret."<br />
<br />
Previous locations for MOP have included IMSA, Rutgers University, West Point (US Military Academy), and the US Naval Academy.<br />
<br />
MOP is not only a training camp but also a competition in and of itself. In addition to the regularly administered practice olympiads and the weekly team contest, returning students write and administer the [[ELMO]] (an amorphous acronym) and the USEMO (the USEless Math Olympiad).<br />
<br />
Popular pastimes at MOP include chess, card games, Mafia, and Ultimate Frisbee. Recently, Mafia was banned due to an incident involving the police; fortunately, nobody cares about the ban.<br />
<br />
== Links ==<br />
*[http://www.unl.edu/amc/a-activities/a6-mosp/mosp.shtml AMC MOP page]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Mathematical_Olympiad_Summer_Program&diff=26564Mathematical Olympiad Summer Program2008-06-16T00:14:12Z<p>MysticTerminator: /* History and Culture */</p>
<hr />
<div>{{WotWAnnounce|week=June 13-19}}<br />
<br />
The '''Mathematical Olympiad Summer Program''' (abbreviated '''MOP''') is a 3-week intensive problem solving camp held at the University of Nebraska-Lincoln to help high school students prepare for math olympaids, especially the [[International Mathematical Olympiad]]. While the program is free to participants, invitations are limited to the top finishers on [[USAMO]].<br />
<br />
== Purpose ==<br />
One purpose of MOP is to select and train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results of the USAMO and the TST are weighted equally when selecting the US IMO team.<br />
<br />
The other important purpose of MOP is to train younger students in Olympiad-level problem solving.<br />
<br />
== Information ==<br />
MOP is currently held at the University of Nebraska-Lincoln. While the dates vary from year to year, MOP is generally held in the last three weeks of June. This year the dates are June 10-July 3, 2008.<br />
<br />
Invitations are extended to the top non-Canadian finishers on USAMO. Students receiving invitations can be divided into four groups:<br />
<br />
USAMO winners: The Americans among the top 12 finishers on USAMO are invited to MOP regardless of their age. Additionally, they are invited to take the Team Selection Test and are viewed as potential members of the American IMO team for that year.<br />
<br />
Top non-senior USAMO finishers: In addition to the winners, the next 18 or so non-senior non-Canadian finishers are invited to attend MOP. This group is viewed as potential IMO team members for future years, although in extreme circumstances (including 2006) IMO team members for that year have been drawn from this pool.<br />
<br />
Top 30 freshman: The top 30 freshman on USAMO are invited to attend MOP with the goal of providing them with a foundation in olympiad level mathematics.<br />
<br />
For 2008, another group has been added. The girls who will be representing the United States at the Chinese Girls Math Olympiad will attend MOP to prepare for that contest.<br />
<br />
== Structure of the Program ==<br />
<br />
MOP is divided into three groups that roughly correspond with the first three kinds of invitations. Black MOP consists of that year's USAMO winners and contains the IMO team members and alternates. Blue MOP is for the second group of invitees and mostly consists of students who just completed their junior or sophomore year of high school, although in exceptional cases some 7th and 8th graders have participated. Finally, Red MOP consists of all the freshman who were invited to participate. Students and instructors have discretion in selecting which group they're part of and may choose to transfer part way through the program; this generally involves members of Black dropping down to Blue or occasionally members of Red promoting themselves to Blue. The three groups take classes and practice tests separately, are given different levels of material to practice with, and to a certain extent are distinct socially. For the most part, Blue and Black are more closely associated to each other than either one is to Red; for example, members of Blue and Black are placed on the same teams for the team contest.<br />
<br />
Each Weekday consists of three instructional sessions: 8:30 AM - 11:30 AM, 1:15 PM - 3:15 PM, and 8:00 PM - 10:00 PM. Classes usually consist of a lecture followed by a problem set. Solutions are often presented by students with the supervision of an instructor.<br />
<br />
Timed and graded olympiad style tests are an integral part of MOP. Every few days, a 4-hour, 3-question test is administered in place of the afternoon lecture, and is graded with comments within 2-3 days.<br />
<br />
Team tests also occur weekly. Students are divided into teams of five, and work on a set of twenty-five problems for approximately half a week. On the day of the contest, the teams present solutions to problems which have not yet been presented, in arbitrary order. The fun starts when all of the easy problems have been taken, and teams resort to certain creative methods in order to solve a problem.<br />
<br />
The combination of these makes MOP an extraordinarily intense experience. One participant at 2007 MOP calculated that by the end of the second week members of Blue MOP had already spent more time in a classroom than most calculus classes do in a year, and by the end of the third week participants had spent 170 hours over 19 days either in class or taking practice test for an average of roughly 9 hours a day of math- and that's before time spent doing problem sets and working on the team contest outside of class is included.<br />
<br />
== History and Culture ==<br />
MOP was created in 1974 as a training camp for the first United States IMO team. <br />
<br />
At the time that MOP was established the official name was simply "Mathematical Olympiad Program", which was the source of the original abbreviation "MOP". At some point, however, the official name was changed to "Mathematical Olympiad Summer Program" and the official abbreviation became "MOSP". Despite this change, participants and alumni almost universally continue to refer to the program as "MOP". Although some administrators continue to use "MOSP" in official documents, students use "MOP" in every setting. One former participant testifies, "Any lost souls using the other appellation are looked upon with pity and regret."<br />
<br />
Previous locations for MOP have included IMSA, Rutgers University, West Point (US Military Academy), and the US Naval Academy.<br />
<br />
MOP is not only a training camp but also a competition in and of itself. In addition to the regularly administered practice oklympiads and the weekly team contest, returning students write and administer the [[ELMO]] (an amorphous acronymn) and the USEMO (the USEless Math Olympiad).<br />
<br />
Popular pastimes at MOP include chess, card games, Mafia, and Ultimate Frisbee. Recently, Mafia was banned due to an incident involving the police; fortunately, nobody cares about the ban.<br />
<br />
== Links ==<br />
*[http://www.unl.edu/amc/a-activities/a6-mosp/mosp.shtml AMC MOP page]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=User:MysticTerminator&diff=26076User:MysticTerminator2008-05-20T17:36:01Z<p>MysticTerminator: New page: Hi I'm MysticTerminator I'm not sure if I get to edit this page at will or what.</p>
<hr />
<div>Hi I'm MysticTerminator I'm not sure if I get to edit this page at will or what.</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Asian_Pacific_Mathematics_Olympiad&diff=26075Asian Pacific Mathematics Olympiad2008-05-20T17:30:34Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=May 20-27}}<br />
The '''Asian Pacific Mathematics Olympiad''' ('''APMO''') is a mathematics olympiad for high school students in countries on and near the Pacific rim. The olympiad started in 1989, and usually occurs on a Monday in March (often around the date of the AIME and many times the day before). It consists of five problems, worth seven points each, to be solved in one four-hour session. Each country largely grades its own papers. A country may have any number of unofficial participants, but only ten students' grades from any country are official. The APMO committee meets each year during the [[IMO]].<br />
<br />
In the United States, usually the previous summer's [[MOP]] participants who have not yet graduated are invited to take the APMO.<br />
<br />
Each country can get a maximum of one gold medal, two silver medals, and three bronze medals.<br />
<br />
== See Also ==<br />
<br />
* [[APMO Problems and Solutions]]<br />
* [http://www.kms.or.kr/competitions/apmo/ Website]<br />
* [http://camel.math.ca/CMS/Competitions/APMO/ Unofficial website]<br />
* [http://www.unl.edu/amc/a-activities/a7-problems/problemUSAMO-IMOarchive.shtml APMO, USAMO, USA TST, and IMO Problems]<br />
* [ftp://ftp.uwm.edu/apmo Some APMO problems (.PS)]<br />
<br />
<br />
[[Category:Mathematics competitions]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=User_talk:MysticTerminator&diff=26074User talk:MysticTerminator2008-05-20T17:26:37Z<p>MysticTerminator: Replacing page with '?'</p>
<hr />
<div>?</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Asian_Pacific_Mathematics_Olympiad&diff=26071Asian Pacific Mathematics Olympiad2008-05-20T15:59:16Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=May 20-27}}<br />
The '''Asian Pacific Mathematics Olympiad''' ('''APMO''') is a mathematics olympiad for high school students in countries on and near the Pacific rim. The olympiad started in 1989, and usually occurs in March (often, in fact, the day before the AIME). It consists of five problems, worth seven points each, to be solved in one four-hour session. Each country largely grades its own papers. A country may have any number of unofficial participants, but only ten students' grades from any country are official. The APMO committee meets each year during the [[IMO]].<br />
<br />
In the United States, usually the previous summer's [[MOP]] participants who have not yet graduated are invited to take the APMO.<br />
<br />
Each country can get a maximum of one gold medal, two silver medals, and three bronze medals. This is kind of silly for example like in that year when everyone got a perfect score or something (2006?).<br />
<br />
Also the grades take like forever to get back.<br />
<br />
== See Also ==<br />
<br />
* [[APMO Problems and Solutions]]<br />
* [http://www.kms.or.kr/competitions/apmo/ Website]<br />
* [http://camel.math.ca/CMS/Competitions/APMO/ Unofficial website]<br />
* [http://www.unl.edu/amc/a-activities/a7-problems/problemUSAMO-IMOarchive.shtml APMO, USAMO, USA TST, and IMO Problems]<br />
* [ftp://ftp.uwm.edu/apmo Some APMO problems (.PS)]<br />
<br />
<br />
[[Category:Mathematics competitions]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Asian_Pacific_Mathematics_Olympiad&diff=26070Asian Pacific Mathematics Olympiad2008-05-20T15:58:30Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=May 20-27}}<br />
The '''Asian Pacific Mathematics Olympiad''' ('''APMO''') is a mathematics olympiad for high school students in countries on and near the Pacific rim. The olympiad started in 1989, and usually occurs in March. It consists of five problems, worth seven points each, to be solved in one four-hour session. Each country largely grades its own papers. A country may have any number of unofficial participants, but only ten students' grades from any country are official. The APMO committee meets each year during the [[IMO]].<br />
<br />
In the United States, usually the previous summer's [[MOP]] participants who have not yet graduated are invited to take the APMO.<br />
<br />
Each country can get a maximum of one gold medal, two silver medals, and three bronze medals. This is kind of silly for example like in that year when everyone got a perfect score or something (2006?).<br />
<br />
Also the grades take like forever to get back.<br />
<br />
== See Also ==<br />
<br />
* [[APMO Problems and Solutions]]<br />
* [http://www.kms.or.kr/competitions/apmo/ Website]<br />
* [http://camel.math.ca/CMS/Competitions/APMO/ Unofficial website]<br />
* [http://www.unl.edu/amc/a-activities/a7-problems/problemUSAMO-IMOarchive.shtml APMO, USAMO, USA TST, and IMO Problems]<br />
* [ftp://ftp.uwm.edu/apmo Some APMO problems (.PS)]<br />
<br />
<br />
[[Category:Mathematics competitions]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Noncommutative_geometry&diff=25928Noncommutative geometry2008-05-09T17:46:40Z<p>MysticTerminator: New page: Noncommutative geometry is the study of geometry/topology on a space where the underlying algebra is noncommutative, so that the order of operations in which one performs actions becomes i...</p>
<hr />
<div>Noncommutative geometry is the study of geometry/topology on a space where the underlying algebra is noncommutative, so that the order of operations in which one performs actions becomes important. I don't actually know much about this besides that for some reason a noncommutative sphere seems to arise in quantum mechanics through analysis of the angular momentum representation of <math>su(2)</math>, so maybe someone else should fill in.</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Mathematical_Olympiad_Summer_Program&diff=24693Mathematical Olympiad Summer Program2008-04-13T19:59:39Z<p>MysticTerminator: /* Naming */</p>
<hr />
<div>The '''Mathematical Olympiad Summer Program''' (abbreviated MOP) is a 3-week intensive problem solving camp held at the University of Nebraska - Lincoln to help high school students prepare for math olympaids, most notably the [[International Mathematical Olympiad]]. The [[USAMO]] winners and honorable mentions, the next 15 or so top scoring non-seniors, as well as the top 30 or so 9th graders on the USAMO, are invited to participate.<br />
<br />
== Purpose ==<br />
One purpose of MOP is to select and train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results of the USAMO and the TST are weighted equally when selecting the US IMO team.<br />
<br />
The other important purpose of MOP is to train younger students in Olympiad-level problem solving.<br />
<br />
== Locations ==<br />
MOP is currently held at University of Nebraska, Lincoln. Previous locations have included IMSA, Rutgers University, West Point (US Military Academy), and the US Naval Academy.<br />
<br />
== Structure ==<br />
Each Weekday consists of three instructional sessions: 8:30 AM - 11:30 AM, 1:15 PM - 3:15 PM, and 8:00 PM - 10:00 PM. Classes usually consist of a lecture followed by a problem set. Solutions are often presented by students with the supervision of an instructor. <br />
Timed and graded olympiad style tests are an integral part of MOP. Every few days, a 4-hour, 3-question test is administered, and is graded with comments within 2-3 days. <br />
Team tests also occur weekly. Students are divided into teams of five, and work on a set of twenty-five problems for approximately half a week. On the day of the contest, the teams present solutions to problems which have not yet been presented, in arbitrary order. The fun starts when all of the easy problems have been taken, and teams resort to certain creative methods in order to solve a problem.<br />
<br />
== Naming ==<br />
The [[AMC]] initially abbreviated the program as "MOP", as the name was just "Mathematical Olympiad Program" at that point, but at some point, the official abbreviation was changed to "MOSP", coinciding with the change of the full name to "Mathematical Olympiad Summer Program". Nevertheless, almost all participants and alumni continue to refer to the program as "MOP" so much so that anyone using the other abbreviation is to be pitied.<br />
<br />
== Results ==<br />
<br />
MOP is not only a training camp but also a competition in and of itself. Various tests involving the team contest, the [[ELMO]] (it is not constant what this stands for but the most popular is e^log Math Olympiad), the USEMO, and the overall MOP tests are administered throughout the program. I don't actually remember too many of the results though so somebody else can fill in, but Hyun-Soo Kim's team won ELMO 2005.<br />
<br />
== Links ==<br />
*[http://www.unl.edu/amc/a-activities/a6-mosp/mosp.html AMC MOP page]<br />
<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Cauchy-Schwarz_Inequality&diff=24616Cauchy-Schwarz Inequality2008-04-08T12:30:48Z<p>MysticTerminator: /* Proof */</p>
<hr />
<div>The '''Cauchy-Schwarz Inequality''' (which is known by other names, including '''Cauchy's Inequality''', '''Schwarz's Inequality''', and the '''Cauchy-Bunyakovsky-Schwarz Inequality''') is a well-known [[inequality]] with many elegant applications.<br />
<br />
== Elementary Form ==<br />
<br />
For any real numbers <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math>,<br />
<center><br />
<math><br />
\left( \sum_{i=1}^{n}a_ib_i \right)^2 \le \left (\sum_{i=1}^{n}a_i^2 \right )\left (\sum_{i=1}^{n}b_i^2 \right )<br />
</math>,<br />
</center><br />
with equality when there exist constants <math>\mu, \lambda </math> not both zero such that for all <math> 1 \le i \le n </math>, <math>\mu a_i = \lambda b_i </math>.<br />
<br />
=== Proof ===<br />
<br />
There are several proofs; we will present an elegant one that does not generalize.<br />
<br />
Consider the vectors <math> \mathbf{a} = \langle a_1, \ldots a_n \rangle </math> and <math> {} \mathbf{b} = \langle b_1, \ldots b_n \rangle </math>. If <math>\theta </math> is the [[angle]] formed by <math> \mathbf{a} </math> and <math> \mathbf{b} </math>, then the left-hand side of the inequality is equal to the square of the [[dot product]] of <math> \mathbf{a} </math> and <math> \mathbf{b} </math>, or <math> \left( ||\mathbf{a}|| \cdot ||\mathbf{b}|| \cos\theta \right)^2 </math>. The right hand side of the inequality is equal to <math> \left( ||\mathbf{a}|| \cdot ||\mathbf{b}|| \right)^2 </math>. The inequality then follows from <math> |\cos\theta | \le 1 </math>, with equality when one of <math> \mathbf{a,b} </math> is a multiple of the other, as desired. Note that this is not actually a proof; rather, this result is used to establish that we can define the angles between two vectors in this way.<br />
<br />
=== Complex Form ===<br />
<br />
The inequality sometimes appears in the following form.<br />
<br />
Let <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math> be [[complex numbers]]. Then<br />
<center><br />
<math><br />
\left| \sum_{i=1}^na_ib_i \right|^2 \le \left (\sum_{i=1}^{n}|a_i^2|\right ) \left (\sum_{i=1}^n |b_i^2|\right )<br />
</math>.<br />
</center><br />
This appears to be more powerful, but it follows immediately from<br />
<center><br />
<math><br />
\left| \sum_{i=1}^n a_ib_i \right| ^2 \le \left( \sum_{i=1}^n |a_i| \cdot |b_i| \right)^2 \le \left(\sum_{i=1}^n |a_i^2|\right)\left( \sum_{i=1}^n |b_i^2|\right )<br />
</math>.<br />
</center><br />
<br />
== General Form ==<br />
<br />
Let <math>V </math> be a [[vector space]], and let <math> \langle \cdot, \cdot \rangle : V \times V \mapsto \mathbb{R} </math> be an [[inner product]]. Then for any <math> \mathbf{a,b} \in V </math>,<br />
<center><br />
<math><br />
\langle \mathbf{a,b} \rangle^2 \le \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle<br />
</math>,<br />
</center><br />
with equality if and only if there exist constants <math>\mu, \lambda </math> not both zero such that <math> \mu\mathbf{a} = \lambda\mathbf{b} </math>.<br />
<br />
=== Proof 1 ===<br />
<br />
Consider the polynomial of <math> t </math><br />
<center><br />
<math><br />
\langle t\mathbf{a + b}, t\mathbf{a + b} \rangle = t^2\langle \mathbf{a,a} \rangle + 2t\langle \mathbf{a,b} \rangle + \langle \mathbf{b,b} \rangle<br />
</math>.<br />
</center><br />
This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., <math> \langle \mathbf{a,b} \rangle^2 </math> must be less than or equal to <math> \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle </math>, with equality when <math> \mathbf{a = 0} </math> or when there exists some scalar <math>-t </math> such that <math> -t\mathbf{a} = \mathbf{b} </math>, as desired.<br />
<br />
=== Proof 2 ===<br />
<br />
We consider<br />
<center><br />
<math><br />
\langle \mathbf{a-b, a-b} \rangle = \langle \mathbf{a,a} \rangle + \langle \mathbf{b,b} \rangle - 2 \langle \mathbf{a,b} \rangle<br />
</math>.<br />
</center><br />
Since this is always greater than or equal to zero, we have<br />
<center><br />
<math><br />
\langle \mathbf{a,b} \rangle \le \frac{1}{2} \langle \mathbf{a,a} \rangle + \frac{1}{2} \langle \mathbf{b,b} \rangle<br />
</math>.<br />
</center><br />
Now, if either <math> \mathbf{a} </math> or <math> \mathbf{b} </math> is equal to <math> \mathbf{0} </math>, then <math> \langle \mathbf{a,b} \rangle^2 = \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle = 0 </math>. Otherwise, we may [[normalize]] so that <math> \langle \mathbf {a,a} \rangle = \langle \mathbf{b,b} \rangle = 1 </math>, and we have<br />
<center><br />
<math><br />
\langle \mathbf{a,b} \rangle \le 1 = \langle \mathbf{a,a} \rangle^{1/2} \langle \mathbf{b,b} \rangle^{1/2}<br />
</math>,<br />
</center><br />
with equality when <math>\mathbf{a} </math> and <math> \mathbf{b} </math> may be scaled to each other, as desired.<br />
<br />
=== Examples ===<br />
<br />
The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the '''Cauchy-Schwarz Inequality for Integrals''': for integrable functions <math> f,g : [a,b] \mapsto \mathbb{R} </math>,<br />
<center><br />
<math><br />
\left( \int_{a}^b f(x)g(x)dx \right)^2 \le \int_{a}^b [f(x)]^2dx \cdot \int_a^b [g(x)]^2 dx<br />
</math>,<br />
</center><br />
with equality when there exist constants <math> \mu, \lambda </math> not both equal to zero such that for <math> t \in [a,b] </math>,<br />
<center><br />
<math><br />
\mu \int_a^t f(x)dx = \lambda \int_a^t g(x)dx<br />
</math>.<br />
</center><br />
<br />
==Problems==<br />
===Introductory===<br />
*Consider the function <math>f(x)=\frac{(x+k)^2}{x^2+1},x\in (-\infty,\infty)</math>, where <math>k</math> is a positive integer. Show that <math>f(x)\le k^2+1</math>. ([[User:Temperal/The_Problem_Solver's Resource Competition|Source]])<br />
===Intermediate===<br />
*Let <math>ABC </math> be a triangle such that<br />
<center><br />
<math><br />
\left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2<br />
</math>,<br />
</center><br />
where <math>s </math> and <math>r </math> denote its [[semiperimeter]] and [[inradius]], respectively. Prove that triangle <math>ABC </math> is similar to a triangle <math>T </math> whose side lengths are all positive integers with no common divisor and determine those integers.<br />
([[2002 USAMO Problems/Problem 2|Source]])<br />
===Olympiad===<br />
*<math>P</math> is a point inside a given triangle <math>ABC</math>. <math>D, E, F</math> are the feet of the perpendiculars from <math>P</math> to the lines <math>BC, CA, AB</math>, respectively. Find all <math>P</math> for which<br />
<br />
<center><br />
<math><br />
\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}<br />
</math><br />
</center><br />
<br />
is least.<br />
<br />
([[1981 IMO Problems/Problem 1|Source]])<br />
<br />
== Other Resources ==<br />
* [http://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality Wikipedia entry]<br />
<br />
===Books===<br />
* [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities] by J. Michael Steele.<br />
* [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities.<br />
<br />
<br />
[[Category:Inequality]]<br />
[[Category:Theorems]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Manifold&diff=24615Manifold2008-04-08T12:29:25Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=March 28-April 5}}<br />
A '''manifold''' is a [[topological space]] locally [[homeomorphic]] to an [[open set | open]] [[ball]] in some [[Euclidean space]]. Informally, this says that if one were living on a point in a manifold, the region surrounding any point would look just like "normal" Euclidean space, i.e. <math>\mathbb{R}^n</math> for some <math>n</math>. For example, the interior of a Mobius strip (that is, excluding its edge) or the surface of an infinite cylinder is a two-dimensional manifold because from each point on either surface the immediate neighborhood is topologically the same as the usual [[Euclidean plane]], even though ''globally'' neither of these surfaces looks much like the plane.<br />
<br />
The [[Whitney Embedding Theorem]] allows us to visualise manifolds as being [[embedding | embedded]] in some Euclidean space. <br />
<br />
Note that the above describes a manifold in the topological category; in the smooth (analytic, holomorphic, etc) category, one would require the patching homeomorphisms to in fact be <math>C^{\infty}</math> (analytic, holomorphic, etc). <br />
<br />
There are also the generalizations of a manifold with boundary, a manifold with corners, and manifolds with even more funky singular points.<br />
<br />
==Definition==<br />
A topological space <math>X</math> is said to be a manifold if and only if <br />
<br />
*<math>X</math> is [[Seperation axioms|Hausdorff]]<br />
<br />
*<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base (topology) | base]].<br />
<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Manifold&diff=24599Manifold2008-04-08T03:19:03Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=March 28-April 5}}<br />
A '''manifold''' is a [[topological space]] locally [[homeomorphic]] to an [[open set | open]] [[ball]] in some [[Euclidean space]]. Informally, this says that if one were living on a point in a manifold, the region surrounding any point would look just like "normal" Euclidean space, i.e. <math>\mathbb{R}^n</math> for some <math>n</math>. For example, the interior of a Mobius strip (that is, excluding its edge) or the surface of an infinite cylinder is a two-dimensional manifold because from each point on either surface the immediate neighborhood is topologically the same as the usual [[Euclidean plane]], even though ''globally'' neither of these surfaces looks much like the plane.<br />
<br />
The [[Whitney Embedding Theorem]] allows us to visualise manifolds as being [[embedding | embedded]] in some Euclidean space. <br />
<br />
Note that the above describes a manifold in the topological category; in the smooth (analytic, holomorphic, etc) category, one would require the patching homeomorphisms to in fact be <math>C^{\infty}</math> (analytic, holomorphic, etc). <br />
<br />
==Definition==<br />
A topological space <math>X</math> is said to be a manifold if and only if <br />
<br />
*<math>X</math> is [[Seperation axioms|Hausdorff]]<br />
<br />
*<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base (topology) | base]].<br />
<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Manifold&diff=24598Manifold2008-04-08T03:17:16Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=March 28-April 5}}<br />
A '''manifold''' is a [[topological space]] locally [[homeomorphic]] to an [[open set | open]] [[ball]] in some [[Euclidean space]]. Informally, this says that if one were living on a point in a manifold, the region surrounding any point would look just like "normal" Euclidean space, i.e. <math>\mathbb{R}^n</math> for some <math>n</math>. For example, the interior of a Mobius strip (that is, excluding its edge) or the surface of an infinite cylinder is a two-dimensional manifold because from each point on either surface the immediate neighborhood is indistinguishable from the usual [[Euclidean plane]], even though ''globally'' neither of these surfaces looks much like the plane.<br />
<br />
The [[Whitney Embedding Theorem]] allows us to visualise manifolds as being [[embedding | embedded]] in some Euclidean space. <br />
<br />
==Definition==<br />
A topological space <math>X</math> is said to be a manifold if and only if <br />
<br />
*<math>X</math> is [[Seperation axioms|Hausdorff]]<br />
<br />
*<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base (topology) | base]].<br />
<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Analysis&diff=24545Analysis2008-04-04T01:41:35Z<p>MysticTerminator: </p>
<hr />
<div>'''Analysis''' is the part of mathematics that primarily deals with continuity, as opposed to discreteness. While there is no way to define exactly what is "analysis" and what is not, one can be pretty sure that something has an analytical flavor every time he hears such words as [[limit]], [[integral]], [[derivative]], [[series]], etc.<br />
<br />
The foundations of mathematical analysis as we know it today were laid in 17-20th centuries starting with the development of integral and differential calculus by Newton and Leibnitz and ending with the Lebesgue theory of measure and integration and functional analysis.<br />
<br />
<br />
{{stub}}<br />
<br />
[[Category:Analysis]]<br />
[[Category:Calculus]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Manifold&diff=24544Manifold2008-04-04T01:35:27Z<p>MysticTerminator: New page: Manifold A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. It has some other properties, like having a countable basis or something, but nobo...</p>
<hr />
<div>Manifold<br />
<br />
A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. It has some other properties, like having a countable basis or something, but nobody really cares about these. You can go ahead and think about manifolds as subspaces of some large Euclidean space anyway, since we can do this by the Whitney embedding theorem.</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Algebraic_geometry&diff=24543Algebraic geometry2008-04-04T01:32:32Z<p>MysticTerminator: </p>
<hr />
<div>'''Algebraic geometry''' is the study of solutions of [[polynomial]] equations by means of [[abstract algebra]], and in particular [[ring theory]]. Algebraic geometry is most easily done over [[algebraically closed]] [[field]]s, but it can also be done more generally over any field or even over [[ring]]s. It is not to be confused with [[analytic geometry]], which is use of coordinates to solve geometrical problems.<br />
<br />
== Affine Algebraic Varieties ==<br />
<br />
One of the first basic objects studied in algebraic geometry is a [[variety]]. Let <math>\mathbb{A}^k</math> denote [[affine]] <math>k</math>-space, i.e. a [[vector space]] of [[dimension]] <math>k</math> over an algebraically closed field, such as the field <math>\mathbb{C}</math> of [[complex number]]s. (We can think of this as <math>k</math>-dimensional "complex Euclidean" space.) Let <math>R=\mathbb{C}[X_1,\ldots,X_k]</math> be the [[polynomial ring]] in <math>k</math> variables, and let <math>I</math> be a [[maximal ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^k\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''.<br />
<br />
== Projective Varieties ==<br />
Let k be a field. A projective variety over k is a projective scheme over k. Projective varieties are algebraic varieties.<br />
<br />
== Schemes ==<br />
Let <math>A</math> be a ring and <math>X=\operatorname{Spec}A</math>. An affine scheme is a ringed topological space isomorphic to some <math>(\operatorname{Spec }A,\mathcal{O}_{\operatorname{Spec}A})</math>.<br />
A scheme is a ringed topological space <math>(X,\mathcal{O}_X)</math> admitting an open covering <math>\{U_i\}_i</math> such that <math>(U_i,\mathcal{O}_{X|U_i})</math> is an affine scheme for every <math>i</math>.<br />
<br />
== Stacks ==<br />
<br />
I don't know what a stack is. Most people I know are scared of them. If you really want to know, check Wikipedia.<br />
<br />
{{stub}}<br />
<br />
[[Category:Algebra]]<br />
[[Category:Geometry]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Multiplicative_function&diff=24504Multiplicative function2008-04-03T04:04:43Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=March 28-April 5}}<br />
<br />
A '''multiplicative function''' <math>f : S \to T</math> is a [[function]] which [[commute]]s with multiplication. That is, <math>S</math> and <math>T</math> must be [[set]]s with multiplication such that <math>f(x\cdot y) = f(x) \cdot f(y)</math> for all <math>x, y \in S</math>, i.e. it preserves the multiplicative structure. A prominent special case of this would be a homomorphism between groups, which preserves the whole group structure (inverses and identity in addition to multiplication).<br />
<br />
Most frequently, one deals with multiplicative functions <math>f : \mathbb{Z}_{>0} \to \mathbb{C}</math>. These functions appear frequently in [[number theory]], especially in [[analytic number theory]]. In this case, one sometimes also defines ''weak multiplicative functions'': a function <math>f: \mathbb{Z}_{>0} \to \mathbb{C}</math> is weak multiplicative if and only if <math>f(mn) = f(m)f(n)</math> for all pairs of [[relatively prime]] [[integer]]s <math>(m, n)</math>.<br />
<br />
Let <math>f(n)</math> and <math>g(n)</math> be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function of <math>n</math> defined by <cmath>\sum_{d|n} f(d) g(\frac{n}{d})</cmath> is also multiplicative; the Mobius inversion formula relates these two quantities.<br />
<br />
Examples in elementary number theory include the identity map, <math>d(n)</math> the number of divisors, <math>\sigma(n)</math> the sum of divisors, <math>\phi(n)</math> the Euler phi function, <math>\tau(n)</math> (I actually forget what this is), <math>\mu(n)</math> the Mobius function, and bunches of other totally awesome stuff.<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Multiplicative_function&diff=24503Multiplicative function2008-04-03T04:02:57Z<p>MysticTerminator: </p>
<hr />
<div>{{WotWAnnounce|week=March 28-April 5}}<br />
<br />
A '''multiplicative function''' <math>f : S \to T</math> is a [[function]] which [[commute]]s with multiplication. That is, <math>S</math> and <math>T</math> must be [[set]]s with multiplication such that <math>f(x\cdot y) = f(x) \cdot f(y)</math> for all <math>x, y \in S</math>, i.e. it preserves the multiplicative structure. A prominent special case of this would be a homomorphism between groups, which preserves the whole group structure (inverses and identity in addition to multiplication).<br />
<br />
Most frequently, one deals with multiplicative functions <math>f : \mathbb{Z}_{>0} \to \mathbb{C}</math>. These functions appear frequently in [[number theory]], especially in [[analytic number theory]]. In this case, one sometimes also defines ''weak multiplicative functions'': a function <math>f: \mathbb{Z}_{>0} \to \mathbb{C}</math> is weak multiplicative if and only if <math>f(mn) = f(m)f(n)</math> for all pairs of [[relatively prime]] [[integer]]s <math>(m, n)</math>.<br />
<br />
Let <math>f(n)</math> and <math>g(n)</math> be multiplicative in the number theoretic sense ("weak multiplicative"). Then the function of <math>n</math> defined by <cmath>\sum_{d|n} f(d) g(\frac{n}{d})</cmath> is also multiplicative.<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Richard_Rusczyk&diff=18338Richard Rusczyk2007-10-18T15:59:10Z<p>MysticTerminator: </p>
<hr />
<div>Richard Rusczyk is the founder of the [[Art of Problem Solving]] website and cowriter with Sandor Lehosczky of the original Art of Problem Solving books. It is possible to understate his awesomeness but only after expending a great deal of effort. There really should be a lot more stuff that goes here, but I don't know that much about him? Check the About tab. <br />
<br />
== External links ==<br />
*[http://en.wikipedia.org/wiki/Richard_Rusczyk Richard Rusczyk on Wikipedia]<br />
*[http://artofproblemsolving.com/Forum/weblog.php?w=1 Rusczyk's blog ]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:AoPS_Community_Awards&diff=13688AoPS Wiki:AoPS Community Awards2007-03-05T05:52:30Z<p>MysticTerminator: /* National Top 12 */</p>
<hr />
<div>This '''AoPS Community Awards''' page is a celebration of the accomplishments of members of the [[AoPS]] community.<br />
<br />
<br />
== IMO Participants and Medalists ==<br />
This is a list of members of the AoPS community who have competed for their country at the [[International Mathematical Olympiad]].<br />
<br />
=== Participants ===<br />
* Zachary Abel (2006) (AoPS assistant instructor)<br />
* Marco Avila (2006)<br />
* Zarathustra Brady (2006)<br />
* Robert Cordwell (2005)<br />
* Sherry Gong (2002, 2003, 2004, 2005)<br />
* Elyot Grant (2005)<br />
* Darij Grinberg (2006)<br />
* Mahbubul Hasan (2005)<br />
* Daniel Kane (AoPS assistant instructor)<br />
* Kiran Kedlaya (1990, 1991, 1992) ([[Art of Problem Solving Foundation]] board member)<br />
* Viktoriya Krakovna (2006)<br />
* Nate Ince (2004) (AoPS assistant instructor)<br />
* Brian Lawrence (2005) ([[WOOT]] instructor)<br />
* Thomas Mildorf (2005) (AoPS assistant instructor)<br />
* Alison Miller (2004) (AoPS assistant instructor)<br />
* Richard Peng (2005, 2006)<br />
* David Rhee (2004, 2005, 2006)<br />
* Peng Shi (2004, 2005, 2006)<br />
* Arnav Tripathy (2006)<br />
* [[Naoki Sato]] (AoPS instructor)<br />
* Yi Sun (2006)<br />
* [[Valentin Vornicu]] (AoPS/MathLinks webmaster)<br />
* Melanie Wood (1998, 1999) ([[WOOT]] instructor)<br />
* Alex Zhai (2006)<br />
* Yufei Zhao (2004, 2005, 2006)<br />
* Tigran Sloyan(2003,2004,2005,2006)<br />
<br />
=== Gold medalists ===<br />
* Zarathustra Brady (2006)<br />
* Robert Cordwell (2005)<br />
* Darij Grinberg (2006)<br />
* Kiran Kedlaya (1990, 1992) ([[Art of Problem Solving Foundation]] board member)<br />
* Brian Lawrence (2005) ([[WOOT]] instructor)<br />
* Thomas Mildorf (2005) (AoPS assistant instructor)<br />
* Alison Miller (2004) (AoPS assistant instructor)<br />
* Arnav Tripathy (2006)<br />
* Yufei Zhao (2005)<br />
<br />
=== Silver medalists ===<br />
* Zachary Abel (2006) (AoPS assistant instructor)<br />
* Sherry Gong (2004, 2005)<br />
* Nate Ince (2004) (AoPS assistant instructor)<br />
* Kiran Kedlaya (1991) ([[Art of Problem Solving Foundation]] board member)<br />
* Viktoriya Krakovna (2006)<br />
* Hyun Soo Kim (2005) (AoPS assistant instructor)<br />
* Richard Peng (2005)<br />
* David Rhee (2006)<br />
* Naoki Sato (AoPS instructor)<br />
* Peng Shi (2006)<br />
* Yi Sun (2006)<br />
* [[Sam Vandervelde]] (1989) ([[WOOT]] instructor)<br />
* Melanie Wood (1998, 1999) ([[WOOT]] instructor)<br />
* Alex Zhai (2006)<br />
* Yufei Zhao (2006)<br />
* Tigran Sloyan(2006)<br />
<br />
=== Bronze medalists ===<br />
* Sherry Gong (2003)<br />
* Elyot Grant (2005)<br />
* Richard Peng (2006)<br />
* [[Naoki Sato]] (AoPS instructor)<br />
* [[Valentin Vornicu]] (AoPS/[[MathLinks]] webmaster)<br />
* Yufei Zhao (2004)<br />
* Tigran Sloyan(2004;2005)<br />
<br />
== IPhO Participants and Medalists ==<br />
This is a list of members of the AoPS community who have competed for their country at the [[International Physics Olympiad]].<br />
=== Participants ===<br />
* Sherry Gong (2006)<br />
* Yi Sun (2004)<br />
<br />
=== Gold Medalists ===<br />
* Yi Sun (2004)<br />
<br />
=== Silver Medalists ===<br />
* Sherry Gong (2006)<br />
<br />
== USAMO ==<br />
The following AoPSers have won the [[United States of America Mathematical Olympiad]] (USAMO). (Note that the definition of "winner" has changed over the years -- currently it is the top 12 scores on the USAMO, but in the past it has been the top 6 or top 8 scores.)<br />
=== Perfect Scorers ===<br />
* Daniel Kane (AoPS assistant instructor)<br />
* Kiran Kedlaya (1991) ([[Art of Problem Solving Foundation]] board member)<br />
* Brian Lawrence (2006) ([[WOOT]] instructor)<br />
<br />
=== Winners ===<br />
* Sherry Gong (2006)<br />
* Yi Han (2006)<br />
* Daniel Kane (AoPS assistant instructor)<br />
* Kiran Kedlaya (1990, 1991, 1992) ([[Art of Problem Solving Foundation]] board member)<br />
* Yakov Berchenko Kogan (2006)<br />
* Brian Lawrence (2005, 2006) ([[WOOT]] instructor)<br />
* Tedrick Leung (2006)<br />
* Richard Mccutchen (2006)<br />
* Albert Ni (2005)<br />
* [[David Patrick]] (1988) (AoPS instructor)<br />
* [[Richard Rusczyk]] (1989) (AoPS founder)<br />
* Peng Shi (2006)<br />
* Yi Sun (2006)<br />
* Arnav Tripathy (2006)<br />
* [[Sam Vandervelde]] (1987, 1989) ([[WOOT]] instructor)<br />
* Melanie Wood (1998, 1999) ([[WOOT]] instructor)<br />
* Alex Zhai (2006)<br />
* Yufei Zhao (2006)<br />
<br />
== Putnam Fellows ==<br />
The top 5 students (including ties) on the collegiate [[Putnam Exam|William Lowell Putnam Competition]] are named Putnam Fellows.<br />
* David Ash (1981, 1982, 1983)<br />
* Daniel Kane (2003, 2004, 2005) (AoPS assistant instructor)<br />
* Kiran Kedlaya (1994, 1995, 1996) ([[AoPS Foundation]] board member)<br />
* Matthew Ince (2005) (AoPS assistant instructor)<br />
* Alexander Schwartz (2000, 2002)<br />
* Jan Siwanowicz (2001) <br />
* Melanie Wood (2002) ([[WOOT]] instructor)<br />
<br />
== Siemens Competition Winners ==<br />
The annual [[Siemens Competition]] (formerly Siemens-Westinghouse) is a scientific research competition.<br />
* Michael Viscardi (1st Individual,2005)<br />
* Lucia Mocz (2nd Team, 2006)<br />
<br />
== Clay Junior Fellows ==<br />
Each year since 2003, the [[Clay Mathematics Institute]] has selected 12 Junior Fellows.<br />
* Thomas Belulovich (2005) (AoPS assistant instructor)<br />
* Atoshi Chowdhury (2003) (AoPS assistant instructor)<br />
* Robert Cordwell (2005)<br />
* Eve Drucker (2003) (AoPS assistant instructor)<br />
* Matthew Ince (2004) (AoPS assistant instructor)<br />
* Nate Ince (2004) (AoPS assistant instructor)<br />
* Hyun Soo Kim (2005) (AoPS assistant instructor)<br />
* Raju Krishnamoorthy (2005)<br />
* Alison Miller (2003) (AoPS assistant instructor)<br />
* Brian Rice (2003) (AoPS assistant instructor)<br />
* Dmitry Taubinski (2005) (AoPS assistant instructor)<br />
* Ameya Velingker (2005)<br />
<br />
<br />
== Perfect AIME Scores ==<br />
Very few students have ever achieved a perfect score on the [[American Invitational Mathematics Examination]] (AIME)<br />
* David Benjamin (2006)<br />
* [[Mathew Crawford]] (1992) (AoPS instructor)<br />
* [[Sandor Lehoczky]] (1990) (AoPS author)<br />
* Tedrick Leung (2006)<br />
* Tony Liu (2006)<br />
* [[Richard Rusczyk]] (1989) (AoPS founder)<br />
* [[Sam Vandervelde]] (1988) ([[WOOT]] instructor)<br />
<br />
== Perfect AMC Scores ==<br />
=== Perfect AMC 12 Scores ===<br />
The [[AMC 12]] is a challenging examination for students in grades 12 and below administered by the [[American Mathematics Competitions]].<br />
* Zachary Abel (2005) (AoPS assistant instructor)<br />
* Ruozhou (Joe) Jia (2003) (AoPS assistant instructor)<br />
* Joel Lewis (2003) <br />
* Jonathan Lowd (2003) (AoPS assistant instructor)<br />
* Thomas Mildorf (2004) (AoPS assistant instructor)<br />
* Alison Miller (2004) (AoPS assistant instructor)<br />
* Albert Ni (2003) (AoPS instructor)<br />
* Ajai Sharma (2004)<br />
* Arnav Tripathy (2006)<br />
<br />
=== Perfect AMC 10 Scores ===<br />
The [[AMC 10]] is a challenging examination for students in grades 10 and below administered by the [[American Mathematics Competitions]].<br />
* Noah Arbesfeld (2006, 2007)<br />
* Yifan Cao (2005)<br />
* Zhou Fan (2005)<br />
* Albert Gu (2007)<br />
* Keone Hon (2005)<br />
* Susan Hu (2005)<br />
* Vincent Le (2006)<br />
* Patricia Li (2005)<br />
* Carl Lian (2007)<br />
* Howard Tong (2005)<br />
* Brent Woodhouse (2006)<br />
<br />
=== Perfect AHSME Scores ===<br />
The [[American High School Mathematics Examination]] (AHSME) was the predecessor of the AMC 12.<br />
* Christopher Chang (1994, 1995, 1996)<br />
* [[Mathew Crawford]] (1994, 1995) (AoPS instructor)<br />
* [[David Patrick]] (1988) (AoPS instructor)<br />
<br />
<br />
== MATHCOUNTS ==<br />
[[MathCounts]] is the premier middle school [[mathematics competition]] in the U.S.<br />
=== National Champions ===<br />
* Ruozhou (Joe) Jia (2000) (AoPS assistant instructor)<br />
* Albert Ni (2002) (AoPS instructor)<br />
* Adam Hesterberg (2003)<br />
* Neal Wu (2005)<br />
* Daesun Yim (2006)<br />
<br />
=== National Top 12 ===<br />
* Ashley Reiter Ahlin (1987) ([[WOOT]] instructor)<br />
* Andrew Ardito (2005, 2006)<br />
* David Benjamin (2004, 2005)<br />
* Nathan Benjamin (2005, 2006)<br />
* Christopher Chang (1991, 1992)<br />
* Andrew Chien (2003)<br />
* Peter Chien (2004)<br />
* Joseph Chu (2004)<br />
* [[Mathew Crawford]] (1990, 1991) (AoPS instructor)<br />
* Brian Hamrick (2006)<br />
* Adam Hesterberg (2002, 2003)<br />
* Ruozhou (Joe) Jia (2000) (AoPS assistant instructor)<br />
* Sam Keller (2006)<br />
* Shaunak Kishore (2003, 2004)<br />
* Kiran Kota (2005)<br />
* Brian Lawrence (2003) ([[WOOT]] instructor)<br />
* Karlanna Lewis (2005)<br />
* Daniel Li (2006)<br />
* Patricia Li (2005)<br />
* Poh-Ling Loh (2000)<br />
* Albert Ni (2002) (AoPS assistant instructor)<br />
* Jason Trigg (2002)<br />
* [[Sam Vandervelde]] (1985) ([[WOOT]] instructor)<br />
* Neal Wu (2005, 2006)<br />
* Rolland Wu (2006)<br />
* Daesun Yim (2006)<br />
* Darren Yin (2002)<br />
* Alex Zhai (2004)<br />
* Mark Zhang (2005)<br />
<br />
=== Masters Round Champions ===<br />
* Christopher Chang (1991)<br />
* Brian Lawrence (2003) ([[WOOT]] instructor)<br />
* Sergei Bernstein (2005)<br />
* Daniel Li (2006)<br />
<br />
=== National Test Champions ===<br />
* [[Mathew Crawford]] (1990) (AoPS instructor)<br />
* Adam Hesterberg (2003)<br />
* Sergei Bernstein (2005)<br />
* Neal Wu (2006)<br />
<br />
== Word Power Challenge ==<br />
The [[Reader's Digest National Word Power Challenge]] tests vocabularies of middle school students. The following members of the AoPS community were scholarship winners at the national contest:<br />
* Billy Dorminy (2005) 2nd place<br />
* Joe Shepherd (2006) 1st place<br />
<br />
==National Vocabulary Championship==<br />
The [[National Vocabulary Championship]] is a multi-stage vocabulary contest for high school students. The following are students who have made it to the National level:<br />
*Billy Dorminy (2007)<br />
*Joe Shepherd (2007)<br />
<br />
== See also ==<br />
* [[Academic competitions]]<br />
* [[Mathematics competitions]]<br />
* [[Mathematics competition resources]]<br />
* [[Academic scholarships]]<br />
<br />
<br />
<br />
[[Category:Art of Problem Solving]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Mathematical_Olympiad_Summer_Program&diff=12075Mathematical Olympiad Summer Program2006-12-30T15:35:43Z<p>MysticTerminator: /* Structure */</p>
<hr />
<div>The '''Mathematical Olympiad Summer Program''' (abbreviated MOP) is a 3-week intensive problem solving camp held at the University of Nebraska - Lincoln to help high school students prepare for math olympaids, most notably the [[International Mathematical Olympiad]]. The [[USAMO]] winners and honorable mentions, the next 15 or so top scoring non-seniors, as well as the top 30 or so 9th graders on the USAMO, are invited to participate.<br />
<br />
== Purpose ==<br />
One purpose of MOP is to select and train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results of the USAMO and the TST are weighted equally when selecting the US IMO team.<br />
<br />
The other important purpose of MOP is to train younger students in Olympiad-level problem solving.<br />
<br />
== Locations ==<br />
MOP is currently held at University of Nebraska, Lincoln. Previous locations have included IMSA, Rutgers University, West Point (US Military Academy), and the US Naval Academy.<br />
<br />
== Structure ==<br />
Each Weekday consists of three instructional sessions: 8:30 AM - 11:30 AM, 1:15 PM - 3:15 PM, and 8:00 PM - 10:00 PM. Classes usually consist of a lecture followed by a problem set. Solutions are often presented by students with the supervision of an instructor. <br />
Timed and graded olympiad style tests are an integral part of MOP. Every few days, a 4-hour, 3-question test is administered, and is graded with comments within 2-3 days. <br />
Team tests also occur weekly. Students are divided into teams of five, and work on a set of twenty-five problems for approximately half a week. On the day of the contest, the teams present solutions to problems which have not yet been presented, in arbitrary order. The fun starts when all of the easy problems have been taken, and teams resort to certain creative methods in order to solve a problem.<br />
<br />
== Naming ==<br />
The [[AMC]] initially abbreviated the program as "MOP," but at some point, the official abbreviation was changed to "MOSP." Nevertheless, almost all participants and alumni continue to refer to the program as "MOP" so much so that any lost souls using the other appellation are looked upon with pity and regret. <br />
<br />
== Results ==<br />
<br />
MOP is not only a training camp but also a competition in and of itself. Various tests involving the team contest, the ELMO (it is not constant what this stands for but the most popular is e^log Math Olympiad), the USEMO, and the overall MOP tests are administered throughout the program. I don't actually remember too many of the results though so somebody else can fill in, but Hyun-Soo Kim's team won ELMO 2005. Yeah.<br />
== Links ==<br />
*[http://www.unl.edu/amc/a-activities/a6-mosp/mosp.html AMC MOP page]<br />
<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Mathematical_Olympiad_Summer_Program&diff=12074Mathematical Olympiad Summer Program2006-12-30T15:32:36Z<p>MysticTerminator: </p>
<hr />
<div>The '''Mathematical Olympiad Summer Program''' (abbreviated MOP) is a 3-week intensive problem solving camp held at the University of Nebraska - Lincoln to help high school students prepare for math olympaids, most notably the [[International Mathematical Olympiad]]. The [[USAMO]] winners and honorable mentions, the next 15 or so top scoring non-seniors, as well as the top 30 or so 9th graders on the USAMO, are invited to participate.<br />
<br />
== Purpose ==<br />
One purpose of MOP is to select and train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results of the USAMO and the TST are weighted equally when selecting the US IMO team.<br />
<br />
The other important purpose of MOP is to train younger students in Olympiad-level problem solving.<br />
<br />
== Locations ==<br />
MOP is currently held at University of Nebraska, Lincoln. Previous locations have included IMSA, Rutgers University, West Point (US Military Academy), and the US Naval Academy.<br />
<br />
== Structure ==<br />
Each Weekday consists of three instructional sessions: 8:30 AM - 11:30 AM, 1:15 PM - 3:15 PM, and 8:00 PM - 10:00 PM. Classes usually consist of a lecture followed by a problem set. Solutions are often presented by students with the supervision of an instructor. <br />
Timed and graded olympiad style tests are an integral part of MOSP. Every few days, a 4-hour, 3-question test is administered, and is graded with comments within 2-3 days. <br />
Team tests also occur weekly. Students are divided into teams of five, and work on a set of twenty-five problems for approximately half a week. On the day of the contest, the teams present solutions to problems which have not yet been presented, in arbitrary order. The fun starts when all of the easy problems have been taken, and teams resort to certain creative methods in order to solve a problem.<br />
<br />
== Naming ==<br />
The [[AMC]] initially abbreviated the program as "MOP," but at some point, the official abbreviation was changed to "MOSP." Nevertheless, almost all participants and alumni continue to refer to the program as "MOP" so much so that any lost souls using the other appellation are looked upon with pity and regret. <br />
<br />
== Results ==<br />
<br />
MOP is not only a training camp but also a competition in and of itself. Various tests involving the team contest, the ELMO (it is not constant what this stands for but the most popular is e^log Math Olympiad), the USEMO, and the overall MOP tests are administered throughout the program. I don't actually remember too many of the results though so somebody else can fill in, but Hyun-Soo Kim's team won ELMO 2005. Yeah.<br />
== Links ==<br />
*[http://www.unl.edu/amc/a-activities/a6-mosp/mosp.html AMC MOP page]<br />
<br />
{{stub}}</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:AoPS_Community_Awards&diff=9845AoPS Wiki:AoPS Community Awards2006-08-23T20:38:56Z<p>MysticTerminator: /* National Top 12 */</p>
<hr />
<div>This '''AoPS Community Awards''' page is a celebration of the accomplishments of members of the [[AoPS]] community.<br />
<br />
<br />
== IMO Participants and Medalists ==<br />
This is a list of members of the AoPS community who have competed for their country at the [[International Mathematical Olympiad]].<br />
<br />
=== Participants ===<br />
* Zachary Abel (2006) (AoPS assistant instructor)<br />
* Marco Avila (2006)<br />
* Zarathustra Brady (2006)<br />
* Robert Cordwell (2005)<br />
* Sherry Gong (2002, 2003, 2004, 2005)<br />
* Elyot Grant (2005)<br />
* Darij Grinberg (2006)<br />
* Mahbubul Hasan (2005)<br />
* Daniel Kane (AoPS assistant instructor)<br />
* Kiran Kedlaya (1990, 1991, 1992) ([[Art of Problem Solving Foundation]] board member)<br />
* Viktoriya Krakovna (2006)<br />
* Nate Ince (2004) (AoPS assistant instructor)<br />
* Brian Lawrence (2005) ([[WOOT]] instructor)<br />
* Thomas Mildorf (2005) (AoPS assistant instructor)<br />
* Alison Miller (2004) (AoPS assistant instructor)<br />
* Richard Peng (2005, 2006)<br />
* David Rhee (2004, 2005, 2006)<br />
* Peng Shi (2004, 2005, 2006)<br />
* Arnav Tripathy (2006)<br />
* [[Naoki Sato]] (AoPS instructor)<br />
* Yi Sun (2006)<br />
* [[Valentin Vornicu]] (AoPS/MathLinks webmaster)<br />
* Melanie Wood (1998, 1999) ([[WOOT]] instructor)<br />
* Alex Zhai (2006)<br />
* Yufei Zhao (2004, 2005, 2006)<br />
<br />
=== Gold medalists ===<br />
* Zarathustra Brady (2006)<br />
* Robert Cordwell (2005)<br />
* Darij Grinberg (2006)<br />
* Kiran Kedlaya (1990, 1992) ([[Art of Problem Solving Foundation]] board member)<br />
* Brian Lawrence (2005) ([[WOOT]] instructor)<br />
* Thomas Mildorf (2005) (AoPS assistant instructor)<br />
* Alison Miller (2004) (AoPS assistant instructor)<br />
* Arnav Tripathy (2006)<br />
* Yufei Zhao (2005)<br />
<br />
=== Silver medalists ===<br />
* Zachary Abel (2006) (AoPS assistant instructor)<br />
* Sherry Gong (2004, 2005)<br />
* Nate Ince (2004) (AoPS assistant instructor)<br />
* Kiran Kedlaya (1991) ([[Art of Problem Solving Foundation]] board member)<br />
* Viktoriya Krakovna (2006)<br />
* Hyun Soo Kim (2005) (AoPS assistant instructor)<br />
* Richard Peng (2005)<br />
* David Rhee (2006)<br />
* Naoki Sato (AoPS instructor)<br />
* Peng Shi (2006)<br />
* Yi Sun (2006)<br />
* [[Sam Vandervelde]] (1989) ([[WOOT]] instructor)<br />
* Melanie Wood (1998, 1999) ([[WOOT]] instructor)<br />
* Alex Zhai (2006)<br />
* Yufei Zhao (2006)<br />
<br />
=== Bronze medalists ===<br />
* Sherry Gong (2003)<br />
* Elyot Grant (2005)<br />
* Richard Peng (2006)<br />
* [[Naoki Sato]] (AoPS instructor)<br />
* [[Valentin Vornicu]] (AoPS/[[MathLinks]] webmaster)<br />
* Yufei Zhao (2004)<br />
<br />
== IPhO Participants and Medalists ==<br />
This is a list of members of the AoPS community who have competed for their country at the [[International Physics Olympiad]].<br />
=== Participants ===<br />
* Sherry Gong (2006)<br />
* Yi Sun (2004)<br />
<br />
=== Gold Medalists ===<br />
* Yi Sun (2004)<br />
<br />
=== Silver Medalists ===<br />
* Sherry Gong (2006)<br />
<br />
== USAMO ==<br />
The following AoPSers have won the [[United States of America Mathematical Olympiad]] (USAMO). (Note that the definition of "winner" has changed over the years -- currently it is the top 12 scores on the USAMO, but in the past it has been the top 6 or top 8 scores.)<br />
=== Perfect Scorers ===<br />
* Daniel Kane (AoPS assistant instructor)<br />
* Kiran Kedlaya (1991) ([[Art of Problem Solving Foundation]] board member)<br />
* Brian Lawrence (2006) ([[WOOT]] instructor)<br />
<br />
=== Winners ===<br />
* Sherry Gong (2006)<br />
* Yi Han (2006)<br />
* Daniel Kane (AoPS assistant instructor)<br />
* Kiran Kedlaya (1990, 1991, 1992) ([[Art of Problem Solving Foundation]] board member)<br />
* Yakov Berchenko Kogan (2006)<br />
* Brian Lawrence (2005, 2006) ([[WOOT]] instructor)<br />
* Tedrick Leung (2006)<br />
* Richard Mccutchen (2006)<br />
* Albert Ni (2005)<br />
* [[David Patrick]] (1988) (AoPS instructor)<br />
* [[Richard Rusczyk]] (1989) (AoPS founder)<br />
* Peng Shi (2006)<br />
* Yi Sun (2006)<br />
* Arnav Tripathy (2006)<br />
* [[Sam Vandervelde]] (1987, 1989) ([[WOOT]] instructor)<br />
* Melanie Wood (1998, 1999) ([[WOOT]] instructor)<br />
* Alex Zhai (2006)<br />
* Yufei Zhao (2006)<br />
<br />
== Putnam Fellows ==<br />
The top 5 students (including ties) on the collegiate [[Putnam Exam|William Lowell Putnam Competition]] are named Putnam Fellows.<br />
* David Ash (1981, 1982, 1983)<br />
* Daniel Kane (2003, 2004, 2005) (AoPS assistant instructor)<br />
* Kiran Kedlaya (1994, 1995, 1996) ([[AoPS Foundation]] board member)<br />
* Matthew Ince (2005) (AoPS assistant instructor)<br />
* Alexander Schwartz (2000, 2002)<br />
* Jan Siwanowicz (2001) <br />
* Melanie Wood (2002) ([[WOOT]] instructor)<br />
<br />
== Siemens Competition Winners ==<br />
The annual [[Siemens Competition]] (formerly Siemens-Westinghouse) is a scientific research competition.<br />
* Michael Viscardi (2005)<br />
<br />
<br />
== Clay Junior Fellows ==<br />
Each year since 2003, the [[Clay Mathematics Institute]] has selected 12 Junior Fellows.<br />
* Thomas Belulovich (2005) (AoPS assistant instructor)<br />
* Atoshi Chowdhury (2003) (AoPS assistant instructor)<br />
* Robert Cordwell (2005)<br />
* Eve Drucker (2003) (AoPS assistant instructor)<br />
* Matthew Ince (2004) (AoPS assistant instructor)<br />
* Nate Ince (2004) (AoPS assistant instructor)<br />
* Hyun Soo Kim (2005) (AoPS assistant instructor)<br />
* Raju Krishnamoorthy (2005)<br />
* Alison Miller (2003) (AoPS assistant instructor)<br />
* Brian Rice (2003) (AoPS assistant instructor)<br />
* Dmitry Taubinski (2005) (AoPS assistant instructor)<br />
* Ameya Velingker (2005)<br />
<br />
<br />
== Perfect AIME Scores ==<br />
Very few students have ever achieved a perfect score on the [[American Invitational Mathematics Examination]] (AIME)<br />
* David Benjamin (2006)<br />
* [[Mathew Crawford]] (1992) (AoPS instructor)<br />
* [[Sandor Lehoczky]] (1990) (AoPS author)<br />
* Tedrick Leung (2006)<br />
* Tony Liu (2006)<br />
* [[Richard Rusczyk]] (1989) (AoPS founder)<br />
* Arnav Tripathy (2006)<br />
* [[Sam Vandervelde]] (1988) ([[WOOT]] instructor)<br />
<br />
== Perfect AMC Scores ==<br />
=== Perfect AMC 12 Scores ===<br />
The [[AMC 12]] is a challenging examination for students in grades 12 and below administered by the [[American Mathematics Competitions]].<br />
* Zachary Abel (2005) (AoPS assistant instructor)<br />
* Ruozhou (Joe) Jia (2003) (AoPS assistant instructor)<br />
* Joel Lewis (2003) <br />
* Jonathan Lowd (2003) (AoPS assistant instructor)<br />
* Thomas Mildorf (2004) (AoPS assistant instructor)<br />
* Alison Miller (2004) (AoPS assistant instructor)<br />
* Albert Ni (2003) (AoPS instructor)<br />
* Ajai Sharma (2004)<br />
* Arnav Tripathy (2006)<br />
<br />
=== Perfect AMC 10 Scores ===<br />
The [[AMC 10]] is a challenging examination for students in grades 10 and below administered by the [[American Mathematics Competitions]].<br />
* Yifan Cao (2005)<br />
* Zhou Fan (2005)<br />
* Keone Hon (2005)<br />
* Susan Hu (2005)<br />
* Vincent Le (2006)<br />
* Patricia Li (2005)<br />
* Howard Tong (2005)<br />
* Noah Arbesfeld (2006)<br />
<br />
=== Perfect AHSME Scores ===<br />
The [[American High School Mathematics Examination]] (AHSME) was the predecessor of the AMC 12.<br />
* Christopher Chang (1994, 1995, 1996)<br />
* [[Mathew Crawford]] (1994, 1995) (AoPS instructor)<br />
* [[David Patrick]] (1988) (AoPS instructor)<br />
<br />
<br />
== MATHCOUNTS ==<br />
[[MathCounts]] is the premier middle school [[mathematics competition]] in the U.S.<br />
=== National Champions ===<br />
* Ruozhou (Joe) Jia (2000) (AoPS assistant instructor)<br />
* Albert Ni (2002) (AoPS instructor)<br />
* Adam Hesterberg (2003)<br />
* Neal Wu (2005)<br />
* Daesun Yim (2006)<br />
<br />
=== National Top 12 ===<br />
* Ashley Reiter Ahlin (1987) ([[WOOT]] instructor)<br />
* Andrew Ardito (2005, 2006)<br />
* David Benjamin (2004, 2005)<br />
* Nathan Benjamin (2005, 2006)<br />
* Christopher Chang (1991, 1992)<br />
* Andrew Chien (2003)<br />
* Peter Chien (2004)<br />
* Joseph Chu (2004)<br />
* [[Mathew Crawford]] (1990, 1991) (AoPS instructor)<br />
* Adam Hesterberg (2002, 2003)<br />
* Ruozhou (Joe) Jia (2000) (AoPS assistant instructor)<br />
* Sam Keller (2006)<br />
* Shaunak Kishore (2003, 2004)<br />
* Kiran Kota (2005)<br />
* Brian Lawrence (2003) ([[WOOT]] instructor)<br />
* Karlanna Lewis (2005)<br />
* Daniel Li (2006)<br />
* Patricia Li (2005)<br />
* Albert Ni (2002) (AoPS assistant instructor)<br />
* Jason Trigg (2002)<br />
* [[Sam Vandervelde]] (1985) ([[WOOT]] instructor)<br />
* Neal Wu (2005, 2006)<br />
* Rolland Wu (2006)<br />
* Daesun Yim (2006)<br />
* Darren Yin (2002)<br />
* Alex Zhai (2004)<br />
* Mark Zhang (2005)<br />
<br />
=== Masters Round Champions ===<br />
* Christopher Chang (1991)<br />
* Brian Lawrence (2003) ([[WOOT]] instructor)<br />
* Daniel Li (2006)<br />
<br />
=== National Test Champions ===<br />
* [[Mathew Crawford]] (1990) (AoPS instructor)<br />
* Adam Hesterberg (2003)<br />
* Neal Wu (2006) <br />
<br />
<br />
== Word Power Challenge ==<br />
The [[Reader's Digest National Word Power Challenge]] tests vocabularies of middle school students. The following members of the AoPS community were scholarship winners at the national contest:<br />
* Billy Dorminy (2005) 2nd place<br />
* Joe Shepherd (2006) 1st place<br />
<br />
<br />
<br />
== See also ==<br />
* [[Academic competitions]]<br />
* [[Mathematics competitions]]<br />
* [[Mathematics competition resources]]<br />
* [[Academic scholarships]]<br />
<br />
<br />
<br />
[[Category:Art of Problem Solving]]</div>MysticTerminatorhttps://artofproblemsolving.com/wiki/index.php?title=Base_numbers&diff=6162Base numbers2006-07-06T17:25:14Z<p>MysticTerminator: /* Common bases */</p>
<hr />
<div>== Introduction ==<br />
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 <center><math>\displaystyle 2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math></center><br />
<br />
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 />
<br />
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 />
<br />
=== Converting between bases ===<br />
==== Converting from base b to base 10 ====<br />
The next natural question is: how do we convert a number from another base into base 10? For example, what does <math>4201_5</math> mean? Just like base 10, the first digit to the left of the decimal place tells us how many <math>5^0</math>'s we have, the second tells us how many <math>5^1</math>'s we have, and so forth. Therefore:<br />
<br />
<center><math>\displaystyle 4201_5 = (4\cdot 5^3 + 2\cdot 5^2 + 0\cdot 5^1 + 1\cdot 5^0)_{10}</math></center><br />
<center><math>=4\cdot 125 + 2\cdot 25 + 1</math></center><br />
<center><math>= 551_{10}</math></center><br />
<br />
From here, we can generalize. Let <math>x=(a_na_{n-1}\cdots a_1a_0)_b</math> be an <math>\displaystyle n</math>-digit number in base <math>b</math>. In our example (<math> 2746_{10}</math>) <math>a_3 = 2, a_2 = 7, a_1 = 4</math> and <math>\displaystyle a_0 = 6 </math>. We convert this to base 10 as follows:<br />
<br />
<center><math> x = (a_na_{n-1}\cdots a_1a_0)_b</math></center><br />
<center><math> = (b^n\cdot a_n + b^{n-1}\cdot a_{n-1}+\cdots + b\cdot a_1 + a_0)_{10}</math></center><br />
<br />
==== Converting from base 10 to base b ====<br />
<br />
It turns out that converting from base 10 to other bases is far harder for us than converting from other bases to base 10. This shouldn't be a suprise, though. We work in base 10 ''all the time'' so we are naturally less comfortable with other bases. Nonetheless, it is important to understand how to convert from base 10 into other bases.<br />
<br />
We'll look at two methods for converting from base 10 to other bases.<br />
<br />
===== Method 1 =====<br />
<br />
Let's try converting 1000 base 10 into base 7. Basically, we are trying to find the solution to the equation<br />
<br />
<center><math> 1000 = a_0 + 7a_1 + 49a_2 + 343a_3+3401a_4+\cdots</math></center><br />
<br />
where all the <math>\displaystyle a_i</math> are digits from 0 to 6. Obviously, all the <math>\displaystyle a_i</math> from <math>a_4</math> and up are 0 since otherwise they will add in a number greater than 1000, and all the terms in the sum are nonnegative. Then, we wish to find the largest <math>a_3</math> such that <math>343a_3</math> does not exceed 1000. Thus, <math> a_3= 2</math> since <math>2a_3=686</math> and <math>3a_3=1029</math>. This leaves us with<br />
<br />
<center><math> 1000 = a_0 + 7a_1 + 49 a_2 + 343(2)\Leftrightarrow 314 = a_0 + 7a_1 + 49 a_2.</math></center><br />
<br />
Using similar reasoning, we find that <math>a_2 = 6</math>, leaving us with<br />
<br />
<center><math>20 = a_0 + 7a_1.</math></center><br />
<br />
We use the same procedure twice more to get that <math>a_1=2</math> and <math>\displaystyle a_0=6</math>.<br />
<br />
Finally, we have that <math>1000_{10}=2626_7</math>.<br />
<br />
An alternative version of method 1 is to find the "digits" <math>a_0,a_1,\dots</math> starting with <math>a_0</math>. Note that <math>a_0</math> is just the [[remainder]] of division of <math>1000</math><br />
by <math>7</math>. So, to find it, all we need to do is to carry out one division with remainder. We have <math>1000:7=142(R6)</math>. How do we find <math>a_1</math>, now? It turns out that all we need to do is to find the remainder of the division of the quotient <math>142</math> by <math>7</math>:<br />
<math>142:7=20(R2)</math>, so <math>a_1=2</math>. Now, <math>20:7=2(R6)</math>, so <math>a_3=6</math>. Finally, <math>2:7=0(R2)</math>, so <math>a_4=2</math>. We may continue to divide beyond this point, of course, but it is clear that we will just get <math>0:7=0(R0)</math> during each step.<br />
<br />
Note that both versions of this method use computations in base <math>10</math>. <br />
<br />
It's often a good idea to double check by converting your answer back into base 10, since this conversion is easier to do. We know that <math>2626_7=343\cdot 2 + 6\cdot 49 + 2\cdot 7 + 6=1000</math>, so we can rest assured we got the right answer.<br />
<br />
===== Method 2 =====<br />
We'll exhibit the second method with the same problem used to exhibit the first method.<br />
<br />
The second method is just like how we converted from other bases into base 10. To do this, we pretend that our standard number system is base 7. In base 7, however, there is no digit 7. So 7 is actually represented as 10! Also, the multiplication rules we know do not hold. For example, <math>\displaystyle 3\cdot 3\neq 9</math> (in base 7). For one, there is no 9 in base 7. Second, we need to go back to the definition of multiplication to fully understand what's happening. Multiplication is a shorthand for repeated addition. So, <math>\displaystyle 3\cdot 3 = 3 + 3 + 3 = 12_7</math>.<br />
<br />
In base 7, we have that 10 (the decimal number 10) is 13. Thus, if we view everything from base 7, we are actually converting <math>1000_{13}</math> to base 10. So, this is just <math>\displaystyle 13^{3}</math>. Remember that we aren't doing this in our regular decimal system, so <math>13^3\neq 2197</math>. Instead, we have to compute <math>13\times 13\times 13</math> as <math>(13\times13)\times 13=202\times 13=2626</math>.<br />
<br />
This method can be ''very'' confusing unless you have a very firm grasp on the notion of number systems. <br />
<br />
== Common bases ==<br />
Commonly used bases are 2, 8, 10 (duh!) and 16. The base doesn't necesarily have to be an integer. There are [[complex base | complex]], [[irrational base | irrational]], [[negative base | negative]], [[improper fractional base | fractional]], and many other kinds of bases. The best known one is [[phinary]], which is base [[phi]]; others include "Fibonacci base" and base negative two.<br />
<br />
=== Binary ===<br />
Binary is base 2. It's a favorite among computer programmers. It has just two digits: <math>0</math> and <math>1</math>.<br />
<br />
===Octal ===<br />
Octal is base 8. It was also quite liked by programmers because the octal representation of numbers is 3 times shorter than the binary one and the conversion from octal to binary and back is very easy (can you guess why?). Besides, 8 is quite close to 10 and less than 10, so to learn doing addition and multiplication in base 8 is not very hard: you can basically count in base 10 with partial conversions to base 8 on the way. Let's multiply <math>12345_8</math> by <math>7_8</math>. <math>5\cdot 7=35_{10}=43_8</math> (to get the last result, just divide <math>35</math> by <math>8</math> with remainder). As usual, we write the last digit <math>3</math> down and keep <math>4</math> in mind. Now, <math>4\cdot 7+4=32_{10}=40_8</math>, so we write down <math>0</math>, getting <math>03</math>, and keeping <math>4</math> in mind. <br />
And so on. The time needed to get the answer <math>111103_8</math> only marginally exceeds the time of decimal multiplication (if you are good in division by 8 with remainder, of course).<br />
<br />
=== Decimal ===<br />
Decimal is base 10. It's the base that everyone knows and loves. Most numbers in the world are written without a specified radix and usually it can just be assumed that they are in base 10. The most commonly used explanation for the origin of base 10 for our number system is the number of fingers we have.<br />
<br />
=== Hexadecimal ===<br />
Hexadecimal is base 16. The digits in hexadecimal are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. One of its common uses is for color charts. Hexadecimal numbers are also used by programmers in the same way as octal numbers, but to learn to count in hexadecimal is harder than in octal.<br />
<br />
== History ==<br />
<br />
Base-10 is an apparently obvious counting system because people have 10 fingers. Historically, different societies utilized other systems. The Native American 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. The Roman system (internal link w/explanation?), 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 />
<br />
== 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>MysticTerminator