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Mathematics Competitions Vol xx No 1 2007

The Art of Problem Solving

David Patrick

David Patrick is Vice President of

AoPS Incorporated. He is the author

of Introduction to Counting & Prob-

ability, a discrete math textbook for

middle and high school students, and is

currently working on the sequel, Inter-

mediate Counting & Probability.Heast

was a United states of america Mathematical Olympiad

winner in 1988, earned his Ph.D. in

Mathematics from MIT in 1997, and

did research in noncommutative alge-

bra.

1Hidue southtory

The Fine art of Trouble Solving (AoPS) website,one established in 2003, has

grown to over 29, 000 members.2 Weste believe that it is the largest website

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

adult specifically for high-ability eye and high school students.

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

loftier schoolhouse students." [three]

2 Forum, Blogs, and Wiki

The centre 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 hash out various math problems

onehttp://www.artofproblemsolving.com

twoAll statistics cited in this article are equally of 4 May 2007.

xix

Mathematics Competitions Vol 20 No 1 2007

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

is free for everyone, 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 post-obit (note that these

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

the titles may include misspellings):

– Maximum of minimum of

ai

one+ north

i=ane i

(in the Olympiad Inequalities forum; the forum is L

A

T

East

Ten-compatible

to allow for mathematical discussion)

– "Another TC problem, whose solution I don't empathise" (in the

Computer Scientific discipline and Informatics forum)

– "baseball game" (in the Eye School forum; the problem nether

give-and-take was a eye-school level problem about baseball game)

– "Nysml" (in the New York local forum; the word was about the

recently-concluded New York Statdue east Mathematics League competition)

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

forum)

– "perfect square in different bases" (in the Loftier School Nuts forum)

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

of under 20 minutes.

Any AoPS member can set upwardly his or her own personal weblog, which is

a personal spider web folio on which the owner can mail service anything he or she

wishes. Although the subjects on the web log are unrestricted, many of the

blog entries discuss mathematics and problem solving. Currently, ovdue eastr

750 AoPS users have offseted blogs.

Finally, AoPS has started a wiki about mathematics and problem

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

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

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

problems and/or problem solving technique, and nonetheless others point to

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

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

20

Mathematics Competitions Vol 20 No 1 2007

–Fer m a t 's La s t Theore yard

– Pascal'southward Triangle Related Problems

– 2007 USAMO Problems

–The Art and Craft of Problem Solvingiii

– Asymptote4

– 1997 AIME Problems

iii 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's

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

A

T

East

10-

uniform and graphics-enabled, to permit mathematical discussion.

AoPS offers three 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 be similar in content to a traditional in-school class, but with a

much greater accent on difficult trouble solving. Second are classes

that are designed as preparation for one of the major U.s.a. mathematics

competitions, such as MATHCOUNTS and the various American

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

that eventually pb to the selection of the U.s.a. IMO squad).

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

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

is to do problem-solving at the IMO level. To requite some indication of the

quality of the WOOT students, in 2005-06 over 90% of the US students

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

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

one of the initial AMC contests, only 430 qualified for the USAMO. [1]

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

Paul Zeitz.

4Asymptote is a L

A

T

East

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

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

21

Mathematics Competitions Vol 20 No i 2007

4 AoPS Foundation and USA Math Talent Search

Divide from the principal AoPS website is the Art of Problem Solving

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

solving education for middle and high schoo50 students in the United

States. The Foundation supports 2 major endeavors.

USA Mathematical Talent Search

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

Hulman Establish of Technology. It has run annually every year since,

and after a number of years of being managed by the United states National

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

Foundation in 2004.

The USAMTS runs during the USA schoolhouse yr (roughly September

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

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

http://www.usamts.org website. Studentsouthward have at to the lowest degree ane full

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

full solutions including proofs. Students are permitted to use whatsoever

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

and computers, but may non consult with teachers or other students.

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

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

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

The issues on Circular i of the 2006-07 USAMTS were:

i. When we perform a 'digit slide' on a number, we move its units

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

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

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

number equals 4 times the number?

5http://www.artofproblemsolving.org

22

Mathematics Competitions Vol 20 No 1 2007

two. 1

(a) In how many different ways canorthward the six empty

circles in the diagram at right be filled in withursday 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?

i

(b) In how many diffehire fashions tin can the seven

empty circles in the diagram at right beast filled

in with the numbers 2 through 8 such that each

number is used once, and each number is either

greater than both its neighbors, or less than both

its neighbors?

three. one

49

(a) An equilateral triangle is divided into 25

coinciding smaller equilateral triangles, every bit

shown. Each of the 21 vertices is labeled

with a number such that for whatever three

sequent vertices on a line segment, their

labels form an arithmetics sequence. The

vertices of the original equilateral triangle

are labeled 1, 4, and 9. Find the sum of the 21 labels.

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

are ntwo smaller coinciding equilateral triangles, and the labels of the

original equilateral triangle are a ,b ,andc .

four. Every bespeak in the airplane is colored either red, green, or blue. Prove

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

vertices are the aforementioned color.

5. ABC D is a tetrahedron such that AB =6,BC =8,AC =Advertisement =

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

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

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

equal volume. Line is the intersection of planes P and Q .Finorthwardd

the length of the portion of that is within ABCD .

23

Mathematics Competitions Vol twenty No one 2007

Local Programs

The AoPS Foundation as well 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, Boulder (Colorado), Charlotte,

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

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

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

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

References

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

Mathematical Competitions, Mathematical Association of America,

2006.

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

Competitions,19 #2, 2006.

[3] M. Matchett Westwardood, "Art of Prob50em Solving: A New Resource for

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

2007.

David Patrick

AoPS Incorporated

Alpine, California, U.s.

Electronic mail: patrick@artofproblemsolving.com

24

... Any interconnections are seen equally tenuous [37] [39] and the output of a complex adaptive social processes. This view is like to Ackoff's [one] concept of 'messes' that argues a mess is a system of problems interacting with each other. Stacey takes this statement a step further by denying organisational objectivity, and views organisational activity equally the complex interaction of people and their e'er-changing definitions of organisational life [23]. ...

... We therefore debate that the pinnacle of an adaptive complex problem solving exercise is to reframe and reorganise solutions until the tensions holding a problem together atomize (Meet [1]). We argue 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 be in disharmonize (See the problem structuring literature Mingers and Rosenhead [30]). ...

... Secondly, it would exist better to explore bug through multiple concepts simultaneously to find richer solutions that better explicate the state of affairs and help actors arrange as opposed to traditional ideas that limit complex bug to paucious unilateral interpretations. Even though this has been effectively covered by the systems movement [1] [nineteen], and others [6], nosotros agree with Ulrich [41] who argues that problem solving is more than methodology choice. We see the need to manage dialectial complex issues through engaging them through multiple concepts in order to deliquesce tensions, not 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 be no problems. Problems are virtually likely a conception of our listen. This means solutions are likewise determined by our conceptions that nosotros can mould and adapt to suit our circumstances. For case, 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 process of making explicit which conception of a problem is beingness used. This paper will re-nowadays the argument that problems and their solutions are merely a conception of our brains and because of this we can change and adapt our thinking to match the evolving circumstances. The implications of this is plant in the way nosotros train people in problem solving, especially as nosotros focus heavily on linearity and not complication, as a method of explaining how people conform their trouble solving ability as function of a adaptive process. The newspaper concludes by arguing that this framework needs to be developed into a more than formal procedure so that the 'reality' of problem solving is better understood. A small analogy of adaptive problem solving is included to help understand the concept.

... Firstly, the standard problem solving process relies on steps offset with what the problem is (Metcalfe, 2005). Secondly, if a definitive construction is available, and so the trouble is not messy (Ackoff, 1978). That is, what is available to be optimised is more than probable something that has known constraints (Liebl, 2002). ...

... This means the roles of multiple actors with different agendas and goals tin can be facilitated through a problem structuring arroyo, in order to come up to the place where 'common involvement' is found. Basadur et al. (2000) calls this procedure 'expanding the pie' (come across also Fisher et al., 1991) to utilize negotiation and facilitation to frame bug in a way that creates a ameliorate interpretation of problems that move actors toward common basis (run across Also Ackoff, 1978). Again, these concerns are well supported past 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 fundamental strategy for SSM as it seeks to create a debate almost change. Ackoff (1978) highlighted the role this could play in dissolving circuitous problems by arguing that the conditions (interpretations) that are given to a problem determine what the problem is believed to be. Checkland (2005) and others accept amplified this by making the role of perception and conceptual framing primal to a problem structuring exercise. ...

  • 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 thought that actors must reach a point where a debate about change, through an adaptation of worldviews, can occur. This paper looks at a field study where actors actively used their declared worldviews against each other to fence for change. Even though this process led to a stalling of the method, an argument is made that there is notwithstanding much to be learned from actors who actively structure disagreement. In detail by studying how this process occurs, we can develop new streams of research into problem framing and methodology use that are currently absent problem structuring enquiry.

... Se comenta en el vocabulario popular que para ser "competente" en Matemática es suficiente dominar operaciones aritméticas y algoritmos de cálculo, cuando el estado platonic 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 Republic of cuba se dirigen a evaluar la preparación académica según los conocimientos, las habilidades e 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. ...

Grooming for Mathematical Olympiads is a infinite for collective construction aimed at the development of competences in teachers and students, motivated by the resolution of problems of a higher difficulty than the standards in university degrees of Cuban Higher Instruction. In the consulted literature, has been plant compilation materials on problem solving techniques focused on what should be done in preparation, but do not provide experiences of how this process should be carried out according to the individualities of the contestants, as well every bit the way to control the level of achievement in their performance. According to the spontaneous nature of this activity 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 piece of work by specific competitions in the contestants and their levels of functioning in the resolution of bug from the training for Olympiads. The results obtained in the application of this musical 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 5 editions of the National Olympiads; placing the University of Holguín at elevation positions of the country in this sphere of bookish performance.

... B. Reznick [10]; quoted from T. Gardiner [2] To warn virtually difficulties involved in the recruitment of future mathematicians, I kickoff with a parable which might look excessively clinical. ...

  • Alexandre V. Borovik

Introduction: what is the purpose of this text? Our meeting Where will the adjacent generation of UK mathematicians come from?will concentrate on the educational activity policy issues arising from our desire to nurture time to come math- ematical talent. However, a brief expect at the plan of the meeting shows that no discussion of what mathematical abilities and talent areis scheduled. I hope that nosotros have a shared understanding sufficient for a meaningful conversation. Nevertheless I believe that some coffee pause chats well-nigh the nature of mathematical abilities and their early manifes- tations in children might be useful. To facilitate an informal discussion of a highly elusive topic, I have decided to offer my notes on mathematical thinking for the attending of the participants of the meeting. At this bespeak, 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 practice not correspond results of any systematic study. The notes are mostly based on my teaching experience in Russian federation in the 1970s and 1980s, in a social and cultural environment 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 united nations 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 infinite for collective construction aimed at the evolution of competences in teachers and students, motivated past the resolution of problems of a higher difficulty than the standards in academy degrees of Cuban Higher Pedagogy. In the consulted literature, has been found compilation materials on problem solving techniques focused on what should be done in training, simply do not provide experiences of how this process should be carried out co-ordinate to the individualities of the contestants, also as the style to control the level of achievement in their performance. 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 Full general Superior Mathematics in the Programs of Technical and Verbal 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 performance in the resolution of problems from the preparation 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 concluding five editions of the National Olympiads; placing the Academy of Holguín at top positions of the country in this sphere of academic performance.

Every bit many business processes are collaborative in nature, process leaders or process managers play a pivotal role designing collaboration processes for organization. To support the blueprint task of creating a new collaborative business procedure, best practices or pattern patterns tin be used as building blocks. For such purposes, a library of design patterns and guidelines would be useful, not only to capture the best practices for different activities in the process in a database, merely to also offer 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 process design, specifically for blueprint efforts post-obit the Collaboration Technology approach.

Problem solving through design of systems and physical artifacts is a professional action with major fiscal significance. Issues in modern society tend to grow more circuitous and intricate, and as a response, systems grow larger. Therefore, pattern increasingly has also go a collaborative task. Design in itself already is challenging but collaboration adds its own challenges into the mix. In this newspaper, nosotros explore the challenges of collaborative design. We approach the inquiry problem through design scientific discipline research framework; we synthesize the knowledge base and adept experiences from the environs to propositions nigh the challenges of collaborative design. By forging theme propositions nosotros lay a basis for design and development of meliorate support for collaborative design.

  • 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 complexity. Thus, many models and scientific methods to resolve urban problems are failed. This study is concerned with proposing of a fuzzy arrangement driven approach for classification and solving urban bug. The proposed study investigated mainly the pick of the inputs and outputs of urban systems for nomenclature of urban problems. In this research, v categories of urban problems, respect to fuzzy system arroyo had been recognized: control, polytely, optimizing, open and conclusion making problems. Grounded Theory techniques were and then applied to analyze the data and develop new solving method for each category. The findings indicate that the fuzzy organisation methods are powerful processes and analytic tools for helping planners to resolve urban complex problems. These tools can exist successful where equally others have failed considering both incorporate or accost uncertainty and risk; complexity and systems interacting with other systems.

Art of Problem Solving: A New Resource for Outstanding Mathematics Students," MAA Focus, 27 #3

  • Thou Matchett

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

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

  • M Matchett Wood

M. Matchett Wood, "Fine art of Trouble 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 #two, 2006.