Mar 30, 2010

New Zealand Maths Olympiad Committee online

http://www.nzamt.org.nz/nzimo/category/notes/

For your amusement (integer sequences)

Tuesday, December 15, 2009 14:59
Posted in category Notes

Neil Sloane, who is the father of the encyclopedia of integer sequences, has produced a paper, which highlights seven of them. There are no problems to solve there (other than some hard open ones!), but lots of interesting stuff. For example, you might like to try and prove the following result before reading it:

Consider the sequence defined as follows a(1) = 1, a(2) = 2, and for n >= 3, a(n) is the smallest positive integer not yet in the sequence such that gcd(a(n), a(n-1)) > 1. Prove that every positive integer eventually appears in the sequence.

Michael

Notes on discrete and polyhedral geometry

Friday, August 28, 2009 10:04
Posted in category Combinatorics, Geometry, Notes

Problems in discrete geometry (i.e. the border between combinatorics and geometry) have made several recent appearances on the IMO (typically, granted as Q6). A nice book by Igor Pak covers much of the important material in this area — the book is aimed at undergaduate and graduate students in maths, but the first few chapters in particular are suitable for Olympiad level students.

As in many cases, the important thing is not the results themselves (though Helly’s theorem is a useful tool in lots of setting) but the “style” of proofs in this area.

Michael

Bertrand’s Postulate

Wednesday, August 19, 2009 8:43
Posted in category Notes, Number Theory

The fact that, for every positive integer n, there is a prime between n and 2n is known as Bertrand’s postulate (which is a bit odd, as it’s a theorem, but anyhow …) It arises occasionally in Olympiad style problems (usually with the note “You may assume Bertrand’s Postulate that …”) Michael Nielsen has a nice post giving an elementary proof at the Polymath wiki.

Michael

Transformational geometry

Tuesday, April 21, 2009 22:50
Posted in category Geometry

Here are the notes from Arkadii Slinko’s Auckland squad lecture last weekend. They present solutions to ten geometric problems — some from contests, some classical.  The common theme is the use of geometric transformations.

Solutions to some of the problems are available, and can be obtained by writing to nzmathsolymp@gmail.com.

Geometric inequalities

Tuesday, April 21, 2009 18:20
Posted in category Geometry

These notes by Arkadii Slinko are a gentle introduction to geometric inequalities, with many nice examples.

Solutions to some of the problems are available, and can be obtained by writing to nzmathsolymp@gmail.com.

Limits, continuity and completeness

Wednesday, April 8, 2009 22:09
Posted in category Algebra

Occasionally, in contest problems, it helps to have a careful understanding of real numbers and real-valued functions.  But what, exactly, is a real number?  These Auckland squad training lecture notes by Heather Macbeth outline some basics.

(Update, 19/4/09:  several errors fixed.)


Mar 22, 2010

Free Online MIT Course Materials | Electrical Engineering and Computer Science | MIT OpenCourseWare

Thank you to MIT for the free online resources:

http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/

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Updated within the past 180 days

  Course # Course Title Term
 
Exams and solutionsMultimedia contentAssignments (no solutions)
6.00 Introduction to Computer Science and Programming Fall 2008
 
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6.001 Structure and Interpretation of Computer Programs Spring 2005
 
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6.002 Circuits and Electronics Spring 2007
 
6.003 Signals and Systems Fall 2003
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6.004 Computation Structures Spring 2009
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6.005 Elements of Software Construction Fall 2008
 
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6.006 Introduction to Algorithms Spring 2008
 
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6.012 Microelectronic Devices and Circuits Fall 2003
 
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6.012 Microelectronic Devices and Circuits Fall 2005
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6.012 Microelectronic Devices and Circuits Spring 2009
 
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6.023J Fields, Forces and Flows in Biological Systems Spring 2007
 
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6.061 Introduction to Electric Power Systems Spring 2007
 
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6.090 Building Programming Experience: A Lead-In to 6.001 January (IAP) 2005
 
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6.092 Java Preparation for 6.170 January (IAP) 2006
 
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6.094 Introduction to MATLAB® January (IAP) 2009
 
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6.095J Computational Biology: Genomes, Networks, Evolution Fall 2005
 
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6.096 Algorithms for Computational Biology Spring 2005
 
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6.096 Introduction to C++ January (IAP) 2009
 
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6.097 Fundamentals of Photonics: Quantum Electronics Spring 2006
 
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6.101 Introductory Analog Electronics Laboratory Spring 2007
 
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6.111 Introductory Digital Systems Laboratory Spring 2006
 
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6.152J Micro/Nano Processing Technology Fall 2005
 
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6.161 Modern Optics Project Laboratory Fall 2005
 
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6.163 Strobe Project Laboratory Fall 2005
 
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6.170 Laboratory in Software Engineering Fall 2005
 
6.171 Software Engineering for Web Applications Fall 2003
 
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6.186 Mobile Autonomous Systems Laboratory January (IAP) 2005
 
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6.189 Multicore Programming Primer January (IAP) 2007
 
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6.189 A Gentle Introduction to Programming Using Python January (IAP) 2008
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6.189 A Gentle Introduction to Programming Using Python January (IAP) 2010
 
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6.270 Autonomous Robot Design Competition January (IAP) 2005
 
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6.338J Applied Parallel Computing (SMA 5505) Spring 2005
 
6.370 Robocraft Programming Competition January (IAP) 2005
 
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6.431 Probabilistic Systems Analysis and Applied Probability Spring 2005
 
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6.521J Quantitative Physiology: Cells and Tissues Fall 2004
 
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6.637 Modern Optics Project Laboratory Fall 2005
 
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6.690 Introduction to Electric Power Systems Spring 2007
 
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6.801 Machine Vision Fall 2004
 
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6.803 The Human Intelligence Enterprise Spring 2002
 
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6.803 The Human Intelligence Enterprise Spring 2006
 
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6.804J Computational Cognitive Science Fall 2004
 
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6.805 Ethics and the Law on the Electronic Frontier Fall 2005
 
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6.833 The Human Intelligence Enterprise Spring 2002
 
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6.833 The Human Intelligence Enterprise Spring 2006
 
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6.837 Computer Graphics Fall 2003
 
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6.857 Network and Computer Security Fall 2003
 
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6.866 Machine Vision Fall 2004
 
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6.895 Computational Biology: Genomes, Networks, Evolution Fall 2005
 
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6.901 Inventions and Patents Fall 2005
 
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6.911 Transcribing Prosodic Structure of Spoken Utterances with ToBI January (IAP) 2006
 
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6.912 Introduction to Copyright Law January (IAP) 2006
 
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6.930 Management in Engineering Fall 2004
 
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6.974 Fundamentals of Photonics: Quantum Electronics Spring 2006
 
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Updated within the past 180 days

  Course # Course Title Term
 
Selected lecture notesAssignments and solutionsExams and solutions
6.041 Probabilistic Systems Analysis and Applied Probability Spring 2005
 
Lecture notesAssignments and solutionsExams and solutionsMultimedia content
6.041 Probabilistic Systems Analysis and Applied Probability Spring 2006
 
Assignments and solutions
6.061 Introduction to Electric Power Systems Spring 2007
 
Selected lecture notes
6.092 Bioinformatics and Proteomics January (IAP) 2005
 
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6.094 Introduction to MATLAB® January (IAP) 2009
 
Selected lecture notesExams (no solutions)
6.095J Computational Biology: Genomes, Networks, Evolution Fall 2005
 
Selected lecture notesAssignments and solutions
6.096 Introduction to C++ January (IAP) 2009
 
Lecture notes
6.097 Fundamentals of Photonics: Quantum Electronics Spring 2006
 
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6.189 Multicore Programming Primer January (IAP) 2007
 
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6.231 Dynamic Programming and Stochastic Control Fall 2008
 
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6.241 Dynamic Systems & Control Fall 2003
 
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6.243J Dynamics of Nonlinear Systems Fall 2003
 
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6.245 Multivariable Control Systems Spring 2004
 
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6.251J Introduction to Mathematical Programming Fall 2002
 
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6.252J Nonlinear Programming Spring 2003
 
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6.253 Convex Analysis and Optimization Spring 2004
 
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6.264J Queues: Theory and Applications Spring 2006
 
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6.301 Solid-State Circuits Spring 2003
 
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6.302 Feedback Systems Spring 2007
 
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6.334 Power Electronics Spring 2007
 
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6.336J Introduction to Numerical Simulation (SMA 5211) Fall 2003
 
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6.339J Numerical Methods for Partial Differential Equations (SMA 5212) Spring 2003
 
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6.432 Stochastic Processes, Detection, and Estimation Spring 2004
 
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6.435 System Identification Spring 2005
 
6.436J Fundamentals of Probability Fall 2005
 
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6.441 Transmission of Information Spring 2003
 
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6.524J Molecular, Cellular and Tissue Biomechanics (BE.410J) Spring 2003
 
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6.541J Speech Communication Spring 2004
 
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6.551J Acoustics of Speech and Hearing Fall 2004
 
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6.581J Foundations of Algorithms and Computational Techniques in Systems Biology Spring 2006
 
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6.630 Electromagnetics Fall 2006
 
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6.632 Electromagnetic Wave Theory Spring 2003
 
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6.635 Advanced Electromagnetism Spring 2003
 
6.637 Optical Signals, Devices, and Systems Spring 2003
 
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6.641 Electromagnetic Fields, Forces, and Motion Spring 2009
 
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6.642 Continuum Electromechanics Fall 2008
 
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6.651J Introduction to Plasma Physics I Fall 2003
 
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6.651J Introduction to Plasma Physics I Fall 2006
 
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6.661 Receivers, Antennas, and Signals Spring 2003
 
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6.685 Electric Machines Fall 2005
 
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6.690 Introduction to Electric Power Systems Spring 2007
 
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6.691 Seminar in Electric Power Systems Spring 2006
 
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6.728 Applied Quantum and Statistical Physics Fall 2006
 
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6.730 Physics for Solid-State Applications Spring 2003
 
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6.772 Compound Semiconductor Devices Spring 2003
 
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6.780 Semiconductor Manufacturing Spring 2003
 
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6.781J Submicrometer and Nanometer Technology Spring 2006
 
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6.801 Machine Vision Fall 2004
 
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6.803 The Human Intelligence Enterprise Spring 2002
 
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6.803 The Human Intelligence Enterprise Spring 2006
 
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6.804J Computational Cognitive Science Fall 2004
 
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6.821 Programming Languages Fall 2002
 
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6.823 Computer System Architecture Fall 2005
 
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6.824 Distributed Computer Systems Engineering Spring 2006
 
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6.825 Techniques in Artificial Intelligence (SMA 5504) Fall 2002
 
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6.826 Principles of Computer Systems Spring 2002
 
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6.827 Multithreaded Parallelism: Languages and Compilers Fall 2002
 
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6.829 Computer Networks Fall 2002
 
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6.830 Database Systems Fall 2005
 
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6.831 User Interface Design and Implementation Fall 2004
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6.833 The Human Intelligence Enterprise Spring 2006
 
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6.838 Algorithms for Computer Animation Fall 2002
 
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6.840J Theory of Computation Fall 2006
 
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6.841J Advanced Complexity Theory Fall 2001
 
6.844 Computability Theory of and with Scheme Spring 2003
 
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6.852J Distributed Algorithms Fall 2005
 
6.854J Advanced Algorithms Fall 2005
 
6.854J Advanced Algorithms Fall 2008
 
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6.857 Network and Computer Security Fall 2003
 
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6.859 Integer Programming and Combinatorial Optimization Fall 2004
 
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6.863J Natural Language and the Computer Representation of Knowledge Spring 2003
 
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6.864 Advanced Natural Language Processing Fall 2005
 
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6.866 Machine Vision Fall 2004
 
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6.867 Machine Learning Fall 2006
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6.977 Semiconductor Optoelectronics: Theory and Design Fall 2002
 
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6.977 Ultrafast Optics Spring 2005
 
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6.978J Communications and Information Policy Spring 2006
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6.982J Teaching College-Level Science and Engineering