https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Googology101&feedformat=atomAoPS Wiki - User contributions [en]2021-01-21T18:56:15ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=AoPS_Wiki:Sandbox&diff=30525AoPS Wiki:Sandbox2009-02-27T22:07:05Z<p>Googology101: </p>
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<div>{{AoPSWiki:Sandbox/header}} <!-- Please do not delete this line --><br />
In the computer world, a '''sandbox''' is a place to test and experiment -- essentially, it's a place to play.<br />
<br />
This is the AoPSWiki Sandbox. Feel free to experiment here.<br />
<br />
Warning: anything you place here is subject to deletion without notice.<br />
<br />
=== Sandbox Area ===<br />
<asy>unitsize(1cm);<br />
defaultpen(fontsize(8));<br />
pair A=(-0.5,0.5), B=(0.5,0.5), C=(0.5,-0.5), D=(-0.5,-0.5);<br />
pair K=(0,1.366), L=(1.366,0), M=(0,-1.366), N=(-1.366,0);<br />
draw(A--N--K--A--B--K--L--B--C--L--M--C--D--M--N--D--A);<br />
label("A",A,SE);<br />
label("B",B,SW);<br />
label("C",C,NW);<br />
label("D",D,NE);</asy><br />
<br />
<asy><br />
import graph;<br />
draw(Circle((0,0),20)); // graph - Circle<br />
</asy><br />
----<br />
<math>\begin{align*}<br />
\dfrac{p+q}{p-q}&=1+\dfrac{2q}{p-q}\\<br />
&= 1+\dfrac{14}{2}\\<br />
&= \boxed{7}<br />
\end{align*}</math><br />
----<br />
Testing answers template:<br />
<math>{{answers<br />
| a = 4<br />
| b = 5<br />
| c = 6<br />
| d = 7<br />
| e = 8<br />
}}</math><br />
<br />
[[User:Googology101|googology101]]<sup>[[User talk:Googology101|talk]] &bull; [[Special:Contribs/Googology101|contribs]]</sup> 22:07, 27 February 2009 (UTC)</div>Googology101https://artofproblemsolving.com/wiki/index.php?title=MATHCOUNTS&diff=29496MATHCOUNTS2009-01-12T22:52:36Z<p>Googology101: Grammar...</p>
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<div>'''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] program that has served millions of middle school students since 1984. Sponsored by the [[CNA Foundation]], [[National Society of Professional Engineers]], the [[National Council of Teachers of Mathematics]], and others, the focus of MATHCOUNTS is on [[mathematical problem solving]]. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.<br />
<br />
== MATHCOUNTS Curriculum ==<br />
MATHCOUNTS curriculum includes [[arithmetic]], [[algebra]], [[counting]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solving]] skills.<br />
<br />
Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.<br />
<br />
==Past Winners==<br />
* 1984: Michael Edwards, Texas<br />
* 1985: Timothy Kokesh, Oklahoma<br />
* 1986: Brian David Ewald, Florida<br />
* 1987: Russell Mann, Tennessee<br />
* 1988: Andrew Schultz, Illinois<br />
* 1989: Albert Kurz, Pennsylvania<br />
* 1990: Brian Jenkins, Arkansas<br />
* 1991: Jonathan L. Weinstein, Massachusetts<br />
* 1992: Andrei C. Gnepp, Ohio<br />
* 1993: Carleton Bosley, Kansas<br />
* 1994: William O. Engel, Illinois<br />
* 1995: Richard Reifsnyder, Kentucky<br />
* 1996: Alexander Schwartz, Pennsylvania<br />
* 1997: Zhihao Liu, Wisconsin<br />
* 1998: Ricky Liu, Massachusetts<br />
* 1999: Po-Ru Loh, Wisconsin<br />
* 2000: Ruozhou Jia, Illinois<br />
* 2001: Ryan Ko, New Jersey<br />
* 2002: Albert Ni, Illinois<br />
* 2003: Adam Hesterberg, Washington<br />
* 2004: Gregory Gauthier, Illinois<br />
* 2005: Neal Wu, Louisiana<br />
* 2006: Daesun Yim, New Jersey (Daesun is a user on AoPS under the usernames [[User:Treething|Treething]] and [[User:Lazarus|Lazarus]])<br />
* 2007: Kevin Chen, Texas (Kevin is a user on AoPS under the username [[User:binonunquineist|binonunquineist]])<br />
* 2008: Darryl Wu, Washington (youngest winner ever, at 11, as well as the first 6th grader to ever even make the National Countdown Round)<br />
<br />
== MATHCOUNTS Competition Structure ==<br />
=== Sprint Round ===<br />
30 problems in 40 minutes. This round is generally made up questions ranging from (relatively) extremely easy to extremely difficult. Some of the difficult problems are only difficult because calculators are not allowed in this round. Like all of the other rounds, it gets progressively harder from the School-level competition to the National-level competition.<br />
<br />
=== Target Round ===<br />
8 problems given 2 at a time. Each set of two problems is given six minutes, students may not go back to previous rounds even if they finish before time is called. Unlike the Sprint and Countdown rounds, use of calculators is permitted, but like all of the other rounds, it gets progressively harder from the School-level competition to the National-level competition.<br />
<br />
=== Team Round ===<br />
10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions). Like all of the other rounds, it gets progressively harder from the School-level competition to the National-level competition.<br />
<br />
=== Countdown Round ===<br />
High scoring individuals compete head-to-head until a champion is crowned. The Countdown round is facilitated differently in the Chapter, State, and National competitions. In the National competitions, it is the round that determines the champion.<br />
<br />
=== Ciphering Round ===<br />
In some states, (most notably Florida) there is an optional Ciphering Round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4 etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.<br />
<br />
====Chapter and State Competitions====<br />
<br />
In the Chapter and State competitions, the Countdown round is not mandatory. However, if it is deemed official by the chapter/state, the following format must be used:<br />
<br />
*The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.<br />
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*The winner of the first round goes up against the 8th place finisher.<br />
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*The winner of the second round goes up against the 7th place finisher.<br />
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This process is continued until the Countdown Round reaches the top four written competitors. Starting then, the first person to get three question correct wins (as opposed to the best-out-of-three rule).<br />
<br />
If the countdown round is unofficial, any format may be used. Single-elimination bracket-style tournaments are common.<br />
<br />
====National Competition====<br />
<br />
At the National competition, there are a couple structural changes to the Countdown Round. The top 12 (not the top 10) written finishers make it to the Countdown Round, and the format is changed from a ladder competition to a single elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first place written finisher to become champion, equalizing the field.<br />
<br />
At the first round and the second round, the first person to correctly answer three questions wins. However, at the semifinals, the rules slightly change - the first person to correctly answer four questions wins.<br />
<br />
=== Masters Round ===<br />
Top students give in-depth explanations to highly challenging problems. This round is optional at the state level competition and is mandatory at the national competition. At nationals the top two on the written and countdown participate.<br />
<br />
=== Scoring and Ranking ===<br />
An individual's score is their total number of correct sprint round answers plus 2 times their total number of correct target round answers. This total is out of a maximum of 30 + 2(8) = 46 points.<br />
<br />
A team's score is the average of the individual scores of its four members plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score.<br />
<br />
== MATHCOUNTS Competition Levels ==<br />
=== School Competition ===<br />
Students vie for the chance to make their school teams. Problems at this level require the least depth of curriculum.<br />
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=== Chapter Competition ===<br />
Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.<br />
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=== State Competition ===<br />
The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members.<br />
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=== National Competition ===<br />
==== Nation Competition Sites ====<br />
For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.<br />
<br />
* The 2009 competition will be held in Orlando, Florida.<br />
* The 2008 competition was held in Denver, Colorado.<br />
* The 2007 competition was held in Fort Worth, Texas.<br />
* The 2006 competition was held in Arlington, Virginia.<br />
* The 2005 competition was held in Detroit, Michigan.<br />
* The 2004 competition was held in Washington, D.C.<br />
* The 2002 and 2003 competitions were held in Chicago, Illinois.<br />
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==== Rewards ====<br />
<br />
Every competitor at the National competition receives a graphing calculator that varies by year - for example, in 2006 it was a TI-84 Plus Silver Edition with the MATHCOUNTS logo on the back. In 2007, MATHCOUNTS took the logo off. In 2008, they gave TI-<math>n</math>spires to everyone. They also give out a laptop and an 8000 dollar scholarship to the winner.<br />
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== MATHCOUNTS Resources ==<br />
=== MATHCOUNTS Books ===<br />
* [http://www.artofproblemsolving.com/Books/AoPS_B_CP_MC.php MATHCOUNTS books] at the [http://www.artofproblemsolving.com/Books/AoPS_B_About.php AoPS Bookstore]<br />
* [[Art of Problem Solving]]'s [http://www.artofproblemsolving.com/Books/AoPS_B_Rec_Middle.php Introductory subject textbooks] are ideal for students preparing for MATHCOUNTS.<br />
<br />
=== MATHCOUNTS Classes ===<br />
* [[Art of Problem Solving]] hosts [http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesP.php#mc MATHCOUNTS preparation classes].<br />
* [[Art of Problem Solving]] hosts many free MATHCOUNTS [[Math Jams]]. [http://www.artofproblemsolving.com/Community/AoPS_Y_Math_Jams.php Math Jam Schedule]. [http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php Math Jam Transcript Archive].<br />
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=== MATHCOUNTS Online ===<br />
* [http://www.mathcounts.org MATHCOUNTS Homepage]<br />
* [[Art of Problem Solving]] hosts a large [http://www.artofproblemsolving.com/Forum/index.php?f=132 MATHCOUNTS Forum] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23209 MATHCOUNTS Coaches Forum].<br />
* [http://mathcounts.saab.org/ Elias Saab's MATHCOUNTS Preparation Homepage]<br />
* [http://www.unidata.ucar.edu/staff/russ/mathcounts/diaz.html The MATHCOUNTS Bible According to Mr. Diaz]<br />
*[http://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MATHCOUNTS Program] by [[Jeff Boyd]], who coached the 2005, 2007, and 2008 National Champion [[Texas MathCounts]] team.<br />
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== What comes after MATHCOUNTS? ==<br />
Give the following competitions a try and take a look at the [[List of United States high school mathematics competitions]].<br />
* [[American Mathematics Competitions]]<br />
* [[American Regions Math League]]<br />
* [[Mandelbrot Competition]]<br />
* [[Mu Alpha Theta]]<br />
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[[Category:Mathematics competitions]]<br />
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== See also ==<br />
* [[List of national MATHCOUNTS teams]]<br />
* [[MATHCOUNTS historical results]]<br />
* [[Mathematics competition resources]]<br />
* [[Math contest books]]<br />
* [[Math books]]<br />
* [[List of United States middle school mathematics competitions]]<br />
* [[List of United States high school mathematics competitions]]<br />
* [http://www.mathcounts.org/webarticles/anmviewer.asp?a=921&z=71 2006 MATHCOUNTS Countdown Video]</div>Googology101https://artofproblemsolving.com/wiki/index.php?title=Factorial&diff=29435Factorial2009-01-11T03:26:32Z<p>Googology101: </p>
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<div>The '''factorial''' is an important function in [[combinatorics]] and [[analysis]], used to determine the number of ways to arrange objects.<br />
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== Definition ==<br />
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The factorial is defined for [[positive integer]]s as <math>n!=n \cdot (n-1) \cdots 2 \cdot 1 = \prod_{i=1}^n i</math>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>.<br />
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== Additional Information ==<br />
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By [[mathematical convention|convention]], <math>0!</math> is given the value <math>1</math>.<br />
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The [[gamma function]] is a generalization of the factorial to values other than [[nonnegative integer]]s.<br />
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==Prime Factorization==<br />
{{main|Prime factorization}}<br />
Since <math>n!</math> is the product of all positive integers not exceeding <math>n</math>, it is clear that it is divisible by all<br />
primes <math>p\le n</math>, and not divisible by any prime <math>p>n</math>. But what is the power of a prime <math>p\le n</math><br />
in the prime factorization of <math>n!</math>? We can find it as the sum of powers of <math>p</math> in all the factors <math>1,2,\dots, n</math>;<br />
but rather than counting the power of <math>p</math> in each factor, we shall count the number of factors divisible by a given power of <math>p</math>. Among the numbers <math>1,2,\dots,n</math>, exactly <math>\left\lfloor\frac n{p^k}\right\rfloor</math> are divisible by <math>p^k</math> (here <math>\lfloor\cdot\rfloor</math> is the [[floor function]]). The ones divisible by <math>p</math> give one power of <math>p</math>. The ones divisible by <math>p^2</math> give another power of <math>p</math>. Those divisible by <math>p^3</math> give yet another power of <math>p</math>. Continuing in this manner gives<br />
<br />
<math>\left\lfloor\frac n{p}\right\rfloor+<br />
\left\lfloor\frac n{p^2}\right\rfloor+<br />
\left\lfloor\frac n{p^3}\right\rfloor+\dots</math><br />
<br />
for the power of <math>p</math> in the prime factorization of <math>n!</math>. The series is formally infinite, but the terms converge to <math>0</math> rapidly, as it is the reciprocal of an [[exponential function]]. For example, the power of <math>7</math> in <math>100!</math> is just<br />
<math>\left\lfloor\frac {100}{7}\right\rfloor+<br />
\left\lfloor\frac {100}{49}\right\rfloor=14+2=16</math><br />
(<math>7^3=343</math> is already greater than <math>100</math>).<br />
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== Uses ==<br />
<br />
The factorial is used in the definitions of [[combinations]] and [[permutations]], as <math>n!</math> is the number of ways to order <math>n</math> distinct objects.<br />
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==Problems==<br />
===Introductory===<br />
*Find the units digit of the sum<br />
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<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath><br />
<br />
<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math><br />
([[2007 iTest Problems/Problem 6|Source]])<br />
===Intermediate===<br />
*Let <math>P </math> be the product of the first <math>100</math> [[positive integer | positive]] [[odd integer]]s. Find the largest integer <math>k </math> such that <math>P </math> is divisible by <math>3^k .</math><br />
([[2006 AIME II Problems/Problem 3|Source]])<br />
===Olympiad===<br />
*Let <math>p_n (k) </math> be the number of permutations of the set <math>\{ 1, \ldots , n \} , \; n \ge 1 </math>, which have exactly <math>k </math> fixed points. Prove that <center><math>\sum_{k=0}^{n} k \cdot p_n (k) = n!</math>.</center><br />
([[1987 IMO Problems/Problem 1|Source]])<br />
<br />
== See Also ==<br />
*[[Combinatorics]]<br />
<br />
[[Category:Combinatorics]]</div>Googology101https://artofproblemsolving.com/wiki/index.php?title=User:Googology101&diff=29430User:Googology1012009-01-11T03:10:26Z<p>Googology101: New page: I can't tell you much about myself, as there isn't much to tell. All I can say is that I love huge numbers and other forms of impractical mathematics. And I stink at FTW. Not much to say.</p>
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<div>I can't tell you much about myself, as there isn't much to tell. All I can say is that I love huge numbers and other forms of impractical mathematics. And I stink at FTW. Not much to say.</div>Googology101