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Mathematics Competitions Vol twenty No 1 2007

The Art of Problem Solving

David Patrick

David Patrick is Vice President of

AoPS Incorporated. He is the writer

of Introduction to Counting & Prob-

power, a discrete math textbook for

middle and high school students, and is

currently working on the sequel, Inter-

mediate Counting & Probability.He

was a USA Mathematical Olympiad

winner in 1988, earned his Ph.D. in

Mathematics from MIT in 1997, and

did enquiry in noncommutative alge-

bra.

1Hidue southtory

The Art of Problem Solving (AoPS) westebsite,i established in 2003, has

grown to over 29, 000 members.ii Weste believe that information technology is the largest website

of its kind in the English-speaking world, with mathematics resource

developed specifically for high-ability eye and loftier school students.

AoPS has been called "a revolution in mathematics grooming for the tiptop

high schoolhouse students." [3]

two Forum, Blogs, and Wiki

The middle of AoPS is the AoPS Forum, which has over 800 ,000 posts

on a variety of topics, mathematical and otherwise. The AoPS Forum

is for students, parents, and teachers to discuss diverse math problems

1http://www.artofproblemsolving.com

twoAll statistics cited in this article are every bit of four May 2007.

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Mathematics Competitions Vol 20 No one 2007

and other topics of involvement to people interested in math. The Forum

is gratis for anybody, and its members are students and teachers from all

different ages, locations, and abilities. In the belatedly afternoon of May 4,

2007, the list of recent discussions include the following (notation that these

are the bodily discussion titles as they appear on the AoPS Forum, so

the titles may include misspellings):

– Maximum of minimum of

ai

ane+ n

i=one i

(in the Olympiad Inequalities forum; the forum is 50

A

T

East

10-compatible

to allow for mathematical discussion)

– "Some other TC problem, whose solution I don't understand" (in the

Figurer Scientific discipline and Informatics forum)

– "baseball" (in the Heart School forum; the trouble nether

word was a center-school level problem about baseball game)

– "Nysml" (in the New York local forum; the discussion westequally nigh the

recently-ended New York Stateastward Mathematics League contest)

– "Displaying the power of a matrix" (in the College Linear Algebra

forum)

– "perfect foursquare in different bases" (in the High School Nuts forum)

All of the posts listed (and many others not listed) occurred in a bridge

of under twenty minutes.

Any AoPS member tin fix his or her own personal blog, which is

a personal web page on which the owner tin can post annihilation he or she

wishes. Although the subjects on the blog are unrestricted, many of the

blog entries discuss mathematics and problem solving. Currently, oveastr

750 AoPS users have started blogs.

Finally, AoPS has started a wiki about mathematics and problem

solving. A wiki is an online encyclopedia that can be edited past anyone.

The AoPS wiki has over 2,000 articles on a variety of topics: some

are related to mathematical ideas and concepts, others are related to

problems and/or problem solving technique, and notwithstanding others betoken to

other resources. To requite a flavor for the types of articles in the AoPS

wiki, a list of some of the most recently created or edited articles includes:

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Mathematics Competitions Vol 20 No 1 2007

–Fer m a t 'southward La s t Theore m

– Pascal's Triangle Related Problems

– 2007 USAMO Problems

–The Fine art and Craft of Problem Solving3

– Asymptote4

– 1997 AIME Problems

3 Online Classes

AoPS runs a number of online classes specifically designed for strong

students in grades 7–12. All of the classes are conducted in AoPS'south

"virtual classroom," an online, moderated chatroom that is L

A

T

E

X-

compatible and graphics-enabled, to permit mathematical give-and-take.

AoPS offers iii different types of classes. First are subject classes

in traditional secondary school math topics: algebra, counting &

probability, geometry, number theory, and trigonometry. These classes

tend to exist similar in content to a traditional in-school course, but with a

much greater emphasis on difficult trouble solving. Second are classes

that are designed as preparation for i of the major US mathematics

competitions, such as MATHCOUNTS and the various American

Mathematics Competitions (AMC) contests (the AMC contests are those

that eventually lead to the selection of the United States IMO team).

Finally, AoPS offers a year-long Westwardorldwide Online Olympiad Training

(WOOT) program, designed for the very best students whose ambition

is to practice trouble-solving at the IMO level. To give some indication of the

quality of the WestOOT students, in 2005-06 over ninety% of the US students

in WOOT qualified for the 2006 USA Mathematical Olympiad, which is

very selective: of thursdaye approximately 230, 000 students in 2006 who wrote

one of the initial AMC contests, only 430 qualified for the USAMO. [one]

3This is a popular problem-solving textbook, authored by former IMO participant

Paul Zeitz.

4Asymptote is a 50

A

T

E

X plug-in for creating high-quality diagrams. The Asymptote

wiki pages on AoPS are considered the "official" wiki pages of the Asymptote pro ject.

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Mathematics Competitions Vol 20 No ane 2007

4 AoPS Foundation and USA Math Talent Search

Split up from the principal AoPS westwardebsite is the Fine art of Problem Solving

Foundation5 . The AoPS Foundation's mission is to promote trouble

solving education for middle and high school students in the United

States. The Foundation supports two major endeavors.

USA Mathematical Talent Search

The USAMTS was founded in 1989 by George Berzsenyi at the Rose-

Hulman Constitute of Technology. It has run annually every yr since,

and after a number of years of being managed by the Us National

Security Bureau, the management of the contest was passed to the AoPS

Foundation in 2004.

The USAMTS runs during the USA school yr (roughly September

through April), and consists of 4 rounds of 5 questions each. The

USAMTS is a "have-home" contest and is run entirely via the

http://www.usamts.org website. Educateesouthward have at least 1 total

calendar month to work on each round of bug, and must write and submit

full solutions including proofs. Students are permitted to use any

available resource to solve the problems, including books, calculators,

and computers, but may not consult with teachers or other students.

Another unique feature of the USAMTSouthward is that students not only receive

numeric scores on their solutions, but as well receive written feedback on

both the correctness and the writing manner of their submitted work.

The problems on Circular 1 of the 2006-07 USAMTS were:

1. When we perform a 'digit slide' on a number, we movement its units

digit to the front of the number. For instance, the result of a 'digit

slide' on 6471 is 1647. What is the smallest positive integer with

4 as its units digit such that the result of a 'digit slide' on the

number equals 4 times the number?

5http://www.artofproblemsolving.org

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Mathematics Competitions Vol 20 No 1 2007

2. ane

(a) In how many different ways can the southix empty

circles in the diagram at right be filled in with the

numbers 2 through 7 such that each number is

used once, and each number is either greater than

both its neighbors, or less than both its neighbors?

ane

(b) In how many diffehire waydue south can thursdaye seven

empty circles in the diagram at right be filled

in with the numbers ii through viii such that each

number is used in one case, and each number is either

greater than both its neighbors, or less than both

its neighbors?

3. i

49

(a) An equilateral triangle is divided into 25

coinciding smaller equilateral triangles, equally

shown. Each of the 21 vertices is labeled

with a number such that for any three

consecutive vertices on a line segment, their

labels form an arithmetic sequence. The

vertices of the original equilateral triangle

are labeled one, 4, and 9. Find the sum of the 21 labels.

(b) Generalize office (a) by finding the sum of the labels when there

are n2 smaller congruent equilateral triangles, and the labels of the

original equilateral triangle are a ,b ,andc .

4. Every point in the aeroplane is colored either red, green, or blueish. Prove

that there exists a rectangle in the aeroplane such that all four of its

vertices are the same colour.

5. ABC D is a tetrahedron such that AB =vi,BC =8,Air conditioning =Advert =

10, and BD =CD = 12. Aeroplane P is parallel to face ABC and

divides the tetrahedron into two pieces of equal volume. Airplane Q

is parallel to face DBC and as well divides ABCD into two pieces of

equal book. Line is the intersection of planes P and Q .Finorthd

the length of the portion of that is inside ABCD .

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Mathematics Competitions Vol 20 No 1 2007

Local Programs

The AoPS Foundation also supports a number of local programs devoted

to mathematics and problem solving education in specific communities

around the United States. These include a number of Math Circles,

including those in San Diego, Stanford, Bedrock (Colorado), Charlotte,

San Jose, Orange County (California), and Albany (New York). The

AoPS Foundation also supports The Teachers' Circle, a new math circle

designed specifically for teachers in the San Francisco Bay area. The

Teachers' Circle is described in greater particular in [2].

References

[1] 2006 Summary of High School Results and Awards ,Ameastwardricanorth

Mathematical Competitions, Mathematical Clan of America,

2006.

[ii] T. Shubin, "Math Circles for Students and Teachers," Mathematics

Competitions,19 #2, 2006.

[three] One thousand. Matchett Wood, "Fine art of Problem Solving: A New Resource for

Outstanding Mathematics Students," MAA Focus ,27 #3, March

2007.

David Patrick

AoPS Incorporated

Alpine, California, USA

Email: patrick@artofproblemsolving.com

24

... Whatever interconnections are seen as tenuous [37] [39] and the output of a complex adaptive social processes. This view is like to Ackoff's [1] concept of 'messes' that argues a mess is a organisation of issues interacting with each other. Stacey takes this statement a stride farther by denying organisational objectivity, and views organisational activity as the complex interaction of people and their ever-changing definitions of organisational life [23]. ...

... Nosotros therefore contend that the pinnacle of an adaptive complex problem solving exercise is to reframe and reorganise solutions until the tensions holding a problem together atomize (Run across [1]). We contend that the problem and its context are interacting perceptual sub-systems that are in conflict, and hence the epistemologies represented [on the nature of the problem] are going to exist in conflict (Come across the problem structuring literature Mingers and Rosenhead [30]). ...

... Secondly, it would exist better to explore issues through multiple concepts simultaneously to find richer solutions that better explain the state of affairs and help actors adapt as opposed to traditional ideas that limit complex problems to paucious unilateral interpretations. Even though this has been effectively covered by the systems move [1] [nineteen], and others [6], we agree with Ulrich [41] who argues that problem solving is more than than methodology choice. We encounter the need to manage dialectial complex problems through engaging them through multiple concepts in order to dissolve tensions, non resolve them (see Ackoff [1] and Mitroff [31] for further discussion). ...

  • Luke Houghton Luke Houghton
  • Mike Metcalfe

What is a problem? Without people there would exist no issues. Issues are virtually likely a conception of our mind. This ways solutions are likewise adamant by our conceptions that we tin can mould and adapt to suit our circumstances. For example, in considering reasonable solutions to world poverty, it needs to be firstly determined whether the situation is due to God'due south Will, Imperialism or a lack of Capitalism. Thus unstructured problem solving becomes a procedure of making explicit which conception of a problem is being used. This newspaper volition re-present the argument that problems and their solutions are only a conception of our brains and because of this nosotros can alter and suit our thinking to match the evolving circumstances. The implications of this is establish in the mode we train people in problem solving, particularly every bit we focus heavily on linearity and non complexity, as a method of explaining how people conform their problem solving power every bit function of a adaptive procedure. The newspaper concludes by arguing that this framework needs to be adult into a more than formal process then that the 'reality' of problem solving is meliorate understood. A small analogy of adaptive trouble solving is included to assist empathise the concept.

... Firstly, the standard problem solving procedure relies on steps beginning with what the problem is (Metcalfe, 2005). Secondly, if a definitive structure is available, and so the problem is not messy (Ackoff, 1978). That is, what is available to exist optimised is more than likely something that has known constraints (Liebl, 2002). ...

... This means the roles of multiple actors with different agendas and goals tin can exist facilitated through a trouble structuring approach, in order to come to the place where 'common interest' is constitute. Basadur et al. (2000) calls this procedure 'expanding the pie' (see besides Fisher et al., 1991) to use negotiation and facilitation to frame problems in a way that creates a improve interpretation of problems that move actors toward mutual basis (run into Likewise Ackoff, 1978). Once more, these concerns are well supported by the trouble structuring literature (Mingers and Rosenhead, 2004). ...

... This leans on the idea that while troublesome and political (Checkland and Scholes, 1990, p. 43), the accommodation of worldviews is a key strategy for SSM equally it seeks to create a contend about modify. Ackoff (1978) highlighted the role this could play in dissolving circuitous problems by arguing that the weather (interpretations) that are given to a trouble determine what the trouble is believed to be. Checkland (2005) and others have amplified this by making the role of perception and conceptual framing cardinal to a problem structuring practise. ...

  • Luke Houghton Luke Houghton

The idea of accommodating worldviews in problem structuring is a common approach across many methodologies. A key assumption of this research is the idea that actors must reach a bespeak where a fence almost alter, through an accommodation of worldviews, tin can occur. This paper looks at a field study where actors actively used their declared worldviews against each other to fence for modify. Even though this procedure led to a stalling of the method, an argument is made that there is still much to exist learned from actors who actively structure disagreement. In particular by studying how this procedure occurs, we can develop new streams of enquiry into problem framing and methodology utilize that are currently absent-minded problem structuring research.

... Se comenta en el vocabulario pop que para ser "competente" en Matemática es suficiente dominar operaciones aritméticas y algoritmos de cálculo, cuando el estado ideal sería discutir el sentido y la aplicación de las ideas matemáticas. Consciente de esta interpretación reducida se reconoce que una vía para desarrollar esta competencia, pudiera ser el desafío que supone la resolución de problemas en las Olimpiadas de Matemática; en especial porque exige un rendimiento especializado y de alto nivel para tener un resultado exitoso (Patrick, 2007). Estos eventos en Cuba se dirigen a evaluar la preparación académica según los conocimientos, las habilidades eastward ingenio de un estudiante; a la vez que estimula el estudio por esta ciencia, impulsa la investigación, promueve la cooperación, favorece los procesos educativos y propicia la creatividad y la capacidad de decisión. ...

Training for Mathematical Olympiads is a space for collective construction aimed at the development of competences in teachers and students, motivated by the resolution of issues of a higher difficulty than the standards in university degrees of Cuban Higher Didactics. In the consulted literature, has been found compilation materials on problem solving techniques focused on what should be done in preparation, just practise not provide experiences of how this process should be carried out co-ordinate to the individualities of the contestants, as well as the fashion to control the level of achievement in their operation. According to the spontaneous nature of this action and the lack of motivation of its personal components are reflected regularities. From a mixed proposal it is proposed to favor the development of specific competences from the General Superior Mathematics in the Programs of Technical and Exact Sciences of the University of Holguín; A methodology is conceived that includes stages, moments and steps for the accomplishment of its objectives, directed to the work past specific competitions in the contestants and their levels of performance in the resolution of problems from the training for Olympiads. The results obtained in the application of this instrument are favorable in terms of relevance and feasibility, based on the increase in the average score of the contestants and the prizes awarded in the last five editions of the National Olympiads; placing the Academy of Holguín at meridian positions of the land in this sphere of academic operation.

... B. Reznick [ten]; quoted from T. Gardiner [ii] To warn almost difficulties involved in the recruitment of future mathematicians, I start with a parable which might wait excessively clinical. ...

  • Alexandre V. Borovik

Introduction: what is the purpose of this text? Our meeting Where will the next generation of UK mathematicians come up from?will concentrate on the education policy issues arising from our want to nurture future math- ematical talent. However, a brief look at the program of the meeting shows that no discussion of what mathematical abilities and talent areis scheduled. I hope that we take a shared understanding sufficient for a meaningful conversation. Still I believe that some coffee suspension chats well-nigh the nature of mathematical abilities and their early on manifes- tations in children might be useful. To facilitate an breezy word of a highly elusive topic, I have decided to offer my notes on mathematical thinking for the attention of the participants of the meeting. At this point, a disclaimer is necessary. I emphasise that I am not a psychologist nor a specialist in educational theory. My notes are highly personal and very subjective. They do not stand for results of any systematic report. The notes are mostly based on my teaching feel in Russia in the 1970s and 1980s, in a social and cultural surround very afar from the modern British landscape.

... Para una amplia discusión del concepto de compensación ver (Bouyssou, 1986; Vansnick, 1986). Ackoff (1978) escribe: "un resultado que se desea en última instancia se llama un "ideal". Si se formula un problema en términos de abordar una solución platonic, se minimizan los riesgos de pasar por alto consecuencias relevantes en la toma de decisiones. ...

Training for Mathematical Olympiads is a space for collective structure aimed at the development of competences in teachers and students, motivated past the resolution of problems of a higher difficulty than the standards in university degrees of Cuban Higher Pedagogy. In the consulted literature, has been constitute compilation materials on trouble solving techniques focused on what should exist done in grooming, but practise non provide experiences of how this process should be carried out according to the individualities of the contestants, as well as the style to control the level of achievement in their performance. Co-ordinate to the spontaneous nature of this activeness and the lack of motivation of its personal components are reflected regularities. From a mixed proposal information technology is proposed to favor the development of specific competences from the General Superior Mathematics in the Programs of Technical and Exact Sciences of the University of Holguín; A methodology is conceived that includes stages, moments and steps for the accomplishment of its objectives, directed to the work by specific competitions in the contestants and their levels of operation in the resolution of problems from the grooming for Olympiads. The results obtained in the application of this instrument are favorable in terms of relevance and feasibility, based on the increase in the average score of the contestants and the prizes awarded in the last five editions of the National Olympiads; placing the Academy of Holguín at top positions of the country in this sphere of academic performance.

As many business processes are collaborative in nature, process leaders or process managers play a pivotal function designing collaboration processes for organization. To support the design chore of creating a new collaborative business process, best practices or blueprint patterns can be used as building blocks. For such purposes, a library of pattern patterns and guidelines would exist useful, not only to capture the best practices for different activities in the process in a database, simply to also offering the users of this database support in selecting and combining such patterns, and in creating the process blueprint. This paper describes the requirements for a tool for pattern based collaboration procedure blueprint, specifically for design efforts following the Collaboration Applied science approach.

Problem solving through pattern of systems and physical artifacts is a professional activity with major financial significance. Problems in modern social club tend to grow more complex and intricate, and as a response, systems grow larger. Therefore, design increasingly has also become a collaborative task. Design in itself already is challenging but collaboration adds its own challenges into the mix. In this paper, nosotros explore the challenges of collaborative design. Nosotros approach the enquiry problem through pattern scientific discipline research framework; nosotros synthesize the knowledge base of operations and expert experiences from the surroundings to propositions well-nigh the challenges of collaborative pattern. Past forging theme propositions we lay a ground for design and development of better support for collaborative blueprint.

  • Giuseppe Munda Giuseppe Munda
  • Edifici B

Resumen: Cualquier problema de decisión social se caracteriza por conflictos entre valores e intereses que compiten y diferentes grupos y comunidades que los representan. Por ejemplo, en la gestión ambiental, las metas de biodiversidad, los objetivos del paisaje, los servicios directos de diferentes entornos como fuentes de recursos y como sumideros de desechos, los significados históricos y culturales que los lugares tienen para las comunidades, las opciones recreativas que proporcionan los entornos, son una fuente de conflicto. Las diferentes dimensiones de valor pueden estar en conflicto entre sí y dentro de sí mismas, y cualquier decisión otorgará diferentes opiniones buenas y malas para los diferentes agentes tanto en forma espacial como temporal. ¿Cómo se deben resolver esos conflictos? A lo largo de los últimos veinte años se han desarrollado y aplicado una variedad de métodos multicriteriales de ayuda a la decisión, con el fin de facilitar la organización de información tanto ecológica como económica, como base para los procesos de toma de decisiones en materia ambiental. Los métodos multicriteriales no asumen la conmensurabilidad de las diferentes dimensiones del problema, ya que no proveen un único criterio de elección, en este sentido, no existe la necesidad de reducir todos los valores en una sola escala (monetaria, energética,…) ayudando a encuadrar y presentar el problema, facilitando el proceso decisor y la obtención de acuerdos políticos. El diseño metodológico aquí presentado, ha permitido identificar, in varios cas os prácticos, los diferentes actores involucrados, describiendo, al mismo tiempo, los problemas de gestión de una forma simultánea tanto en el riguroso lenguaje científico como en términos socio-políticos. Esto ha permitido delimitar los conflictos sociales y mostrar diferentes posibilidades para su solución a través de compromisos, cooperación y dialogo entre las partes, dando oportunidad a que emergieran soluciones.

Urban problems are problems of organized complication. Thus, many models and scientific methods to resolve urban problems are failed. This report is concerned with proposing of a fuzzy arrangement driven arroyo for classification and solving urban issues. The proposed study investigated mainly the selection of the inputs and outputs of urban systems for nomenclature of urban problems. In this research, 5 categories of urban problems, respect to fuzzy system arroyo had been recognized: control, polytely, optimizing, open and determination making bug. Grounded Theory techniques were then practical to analyze the information and develop new solving method for each category. The findings indicate that the fuzzy system methods are powerful processes and analytic tools for helping planners to resolve urban circuitous problems. These tools tin exist successful where as others have failed because both comprise or address doubtfulness and risk; complication and systems interacting with other systems.

Fine art of Problem Solving: A New Resource for Outstanding Mathematics Students," MAA Focus, 27 #three

  • Thou Matchett

Yard. Matchett Wood, "Art of Trouble Solving: A New Resource for Outstanding Mathematics Students," MAA Focus, 27 #3, March 2007. David Patrick AoPS Incorporated Alpine, California, U.s. Email: patrick@artofproblemsolving.com 24

Art of Trouble Solving: A New Resource for Outstanding Mathematics Students

  • Thou Matchett Woods

M. Matchett Wood, "Art of Problem Solving: A New Resource for Outstanding Mathematics Students," MAA Focus, 27 #3, March 2007.

Math Circles for Students and Teachers

  • T Shubin

T. Shubin, "Math Circles for Students and Teachers," Mathematics Competitions, 19 #ii, 2006.