The Enigma Project aims to inspire interest in mathematics, science and history through interactive presentations and hands-on workshops focusing on codes and cryptography. Presentations include demonstrations of a real WWII Enigma cipher machine, loaned to the project by Simon Singh.
Enigma Project visits are suitable for pupils of all ages and abilities from KS2-KS5. It is primarily aimed at school students, but visits can be organised for any educational establishment. The Enigma Projec t is also suitable for masterclasses, summer schools, Gifted and Talented workshops, teachers' conferences and weekend and evening events - with organisation and activities adjusted as appropriate
About the format
The Enigma Project is split into two parts. At the start of the day all students involved will see the opening presentation together. Throughout the course of the day, each student will be then involved in a codebreaking workshop in a smaller class-sized group.
The opening 50-60min interactive presentation introduces students to cryptography - the science and mathematics of codes and code breaking. Pupils meet various ciphers that have been used throughout history from Ancient Greece to modern day cryptography. Presentations involve audience participation and all students see a genuine WW2 Enigma machine in action, find out how it worked,and discover why it is one of the most infamous cipher machines of all time
Students then get the chance to put their problem solving and logical reasoning skills to the test by taking part in a circus of hands-on code breaking activities. The code breaking workshops last for 50-60min with class-sized groups of pupils working in pairs to crack cryptic messages using a variety of traditional and modern methods from Caesar shift ciphers to ISBN numbers.
Thursday, August 5, 2010
Public and Schools Lectures
Details of our programme of free public and schools lectures are posted below - we update this list and add new lectures regularly. If you would like to receive an email notification about future lectures and events please join our MMP events mailing list.
If you enjoy our lectures, you might also want to listen to the Plus podcast.
Details of future events will be posted here shortly.
Download a map and directions to the Centre for Mathematical Sciences, where the lectures are held, unless marked otherwise, or see the University of Cambridge's online map. The main entrance to the Centre for Mathematical Sciences is via the footpath off Clarkson Road (running next to the Isaac Newton Institute car park) and then through the CMS gatehouse. This is signposted.
If you enjoy our lectures, you might also want to listen to the Plus podcast.
Details of future events will be posted here shortly.
Download a map and directions to the Centre for Mathematical Sciences, where the lectures are held, unless marked otherwise, or see the University of Cambridge's online map. The main entrance to the Centre for Mathematical Sciences is via the footpath off Clarkson Road (running next to the Isaac Newton Institute car park) and then through the CMS gatehouse. This is signposted.
Millennium Mathematics Project
The Millennium Mathematics Project (MMP) is a maths education initiative for ages 5 to 19 and the general public, based at the University of Cambridge and active nationally and internationally. We aim to support maths education and promote the development of mathematical skills and understanding, particularly through enrichment activities. More broadly, we want to help everyone share in the excitement and understand the importance of mathematics.
The project consists of a family of complementary programmes, each of which has a particular focus (download an overview of our activities here):
* the NRICH website - thousands of free resources designed to develop problem solving skills and subject knowledge
* the Plus website - an online magazine opening a door to the world of maths, including a careers library
* the Motivate video-conferencing programme - linking schools to professional mathematicians and scientists to engage in investigative project work
* visits to schools all over the UK and abroad by the Hands-On Maths Roadshow, Enigma Project, Risk and Probability Show and NRICH staff
* the East of England Further Mathematics Support Programme - teaching, support and promotion of Further Maths A-level
* popular mathematics lectures for schools and the general public, held in Cambridge
* the STIMULUS programme - placing Cambridge student volunteers in local schools to assist with maths and science classes
The MMP’s various programmes have won many awards and our resources have been repeatedly commended by the UK Government’s Department for Children, Schools and Families (formerly the Department for Education and Skills). Our web-based mathematical resources attract more than 2.3 million visitors worldwide, and around 30,000 pupils and teachers annually are involved in our hands-on activities. In February 2006 the Queen presented the project with the Queen’s Anniversary Prize for Higher and Further Education (the counterpart to the Queen’s Award for Industry), honouring ‘outstanding achievement and excellence’ at world-class level.
The project consists of a family of complementary programmes, each of which has a particular focus (download an overview of our activities here):
* the NRICH website - thousands of free resources designed to develop problem solving skills and subject knowledge
* the Plus website - an online magazine opening a door to the world of maths, including a careers library
* the Motivate video-conferencing programme - linking schools to professional mathematicians and scientists to engage in investigative project work
* visits to schools all over the UK and abroad by the Hands-On Maths Roadshow, Enigma Project, Risk and Probability Show and NRICH staff
* the East of England Further Mathematics Support Programme - teaching, support and promotion of Further Maths A-level
* popular mathematics lectures for schools and the general public, held in Cambridge
* the STIMULUS programme - placing Cambridge student volunteers in local schools to assist with maths and science classes
The MMP’s various programmes have won many awards and our resources have been repeatedly commended by the UK Government’s Department for Children, Schools and Families (formerly the Department for Education and Skills). Our web-based mathematical resources attract more than 2.3 million visitors worldwide, and around 30,000 pupils and teachers annually are involved in our hands-on activities. In February 2006 the Queen presented the project with the Queen’s Anniversary Prize for Higher and Further Education (the counterpart to the Queen’s Award for Industry), honouring ‘outstanding achievement and excellence’ at world-class level.
National mathematics competitions
Albania
* Olimpiada Kombëtare e Matematikës dhe e Formimit të Përgjithshëm.
* Olimpiada Kombëtare e Matematikës
* Konkursi i Pranverës 1(BMO TST)
* Konkursi i Pranverës 2(IMO TST)
[edit] Argentina
* OMA (Olimpíada Matemática Argentina, http://www.oma.org.ar)
* Olimpiada Matematica Ñandu
* Certamen "El numero de oro"
* Torneo Computacion y Matematicas
[edit] Australia
* APSMO (Australasian Problem Solving Mathematical Olympiads, http://www.apsmo.info)
* AMC (Australian Mathematics Competition, http://www.amt.edu.au)
* AMO (Australian Mathematical Olympiad, http://www.amt.edu.au)
* AIMO (Australian Intermediate Mathematical Olympiad)
* UNSW School Mathematics Competition (http://www.maths.unsw.edu.au/highschool/unsw/highcomps.html)
* HSFOL (Have Sum Fun Online) - an online team competition from the Mathematics Association of West Australia (http://www.havesumfunonline.com/)
[edit] Austria
* ÖMO (Österreichische Mathematik Olympiade) (http://www.oemo.at)
[edit] Belgium
French-speaking students can compete in the OMB (Olympiade Mathématique Belge) consisting of three categories:
* Mini (grades 7 and 8)
* Mide (grades 9 and 10)
* Maxi (grades 11 and 12)
Dutch-speaking students can compete in the VWO (Vlaamse Wiskunde Olympiade), with two categories:
* Junior Wiskunde Olympiade (grades 9 and 10)
* Vlaamse Wiskunde Olympiade (grades 11 and 12)
[edit] Brazil
There are two national competitions in Brazil: the oldest one, OBM, dates back to 1979 and is open to all students, from fifth grade to university.
The other one, OBMEP, was created in 2005 and is open to public school students from fifth grade to high school. In 2008 counted with the participation of 18,3 million students on the first round.
* OBM (Olimpíada Brasileira de Matemática, http://www.obm.org.br/)
* OBMEP (Olimpíada Brasileira de Matemática das Escolas Públicas, http://www.obmep.org.br/)
There are also many regional competitions, usually open to all students of a given state.
* OPM (Olimpíada Paulista de Matemática, http://www.opm.mat.br/)
* OMERJ (Olimpíada de Matemática do Estado do Rio de Janeiro, http://www.omerj.com.br/)
[edit] Bulgaria
* Bulgarian Competition in Mathematics and Informatics (http://www.math.bas.bg/bcmi/)
[edit] Canada
National competitions hosted by Mathematica - The Mathematics Contest Centre:
Multiple Choice:
* Newton Contest (9th grade students)
* Lagrange Contest (8th grade students)
* Euler Contest (7th grade students)
* Pythagoras Contest (6th grade students)
* Fibonacci Contest (5th grade students)
* Byron-Germain Contest (4th grade students)
* Thales Contest (3rd grade students)
National competitions hosted by The Centre for Education in Mathematics and Computing:
Full Solutions:
* Euclid (12th grade students)
* Hypatia (11th grade students)
* Galois (10th grade students)
* Fryer (9th grade students)
Multiple Choice:
* Fermat (11th grade students)
* Cayley (10th grade students)
* Pascal (9th grade students)
* Gauss (7th and 8th grade students)
Other competitions hosted by The Canadian Mathematical Society:
* Canadian Open Mathematics Challenge
* Canadian Mathematics Olympiad, the official mathematical Olympiad of Canada and its International Mathematical Olympiad TST, or team selection test.
MATHChallengers (formerly MathCounts BC) is called MathChallengers since 2005. It is hosted by APEGBC.
[edit] China
* CMO (China Mathematics Olympiad 中国数学奥林匹克)
* CWMO (Western China Mathematics Olympiad)
* CGMO (China Girl Mathematics Olympiad) — for female secondary students
* CSMO (South-eastern China Mathematics Olympiad) — for secondary 4 students
* CNMO (China Northern Mathematics Olympiad)
* National high school math League 全国高中数学联赛
* QAIS (Qingdao American International School 青岛美国国际学校) stock market trading, bridge building and siege engines Grades 7 - 12
[edit] Colombia
* OCM (Colombian Mathematics Olympiad)
* OCMU (Colombian Mathematics Olympiad university)
Web site: http://olimpia.uan.edu.co/olimpiadas/public/frameset.jsp
[edit] Cyprus
* Cyprus Mathematical Olympiad — Mathematics competitions for students in every grade.
* Junior high-school (Gymnasium) Competitions — For students under 15.5 years old (1st, 2nd, and 3rd grade of Gymnasium)
* High School (Lyceum) Competitions — For students above 15.5 years old (1st, 2nd, and 3rd grade of Lyceum)
* European Kangourou Mathematics Competition
[edit] Czech Republic
* MO (Czech National Mathematics Olympiad, http://www.math.muni.cz/~rvmo)
* PraSe (Prague Mathematics Correspondence Seminar for Secondary School Students, http://mks.mff.cuni.cz)
* Brkos (Brno Mathematical Correspondence Seminar for Secondary School Students, http://www.math.muni.cz/~brkos)
[edit] Denmark
* Georg Mohr (Contest for Danish high school students, http://www.georgmohr.dk)
[edit] Germany
* DeMO (Deutsche Mathematik-Olympiade / German Math Olympiad, http://www.mathematik-olympiaden.de)
* BWM (Bundeswettbewerb Mathematik / Federal Math Competition , http://www.bundeswettbewerb-mathematik.de),
* AIMO (Auswahlwettbewerb zur Internationalen Mathematik-Olympiade / Team selection tests for the IMO, http://www.bundeswettbewerb-mathematik.de/imo/main.htm)
* LWMB (Landeswettbewerb Mathematik Bayern / Bavarian Math Competition, http://lwmb.de), LWM (Landeswettbewerb Mathematik Baden-Württemberg / Math Competition of Baden-Württemberg, http://landeswettbewerb-mathematik.de)
[edit] Greece
* Θαλής (Thales) - first round
* Ευκλείδης (Euclides) - second round
* Αρχιμήδης (Archimedes) - third round
* Λευκοπούλειος Διαγωνισμός Πιθανοτήτων και Στατιστικής - "Leukopouleios" Contest on Probabilities And Statistics - a competition that is not related to IMO, organised by ESI (National Statistic Institut)
* Math Kangaroo Competition
Seen also Hellenic Mathematical Society
[edit] Hong Kong
* Hong Kong Mathematics Olympiad
* The Hong Kong Mathematical High Achievers Selection Contest. Only Form 1 to form 3 students may participate in this contest.
* International Mathematical Olympiad Preliminary Selection Contest - Hong Kong (IMO HK Prelim, Official Website)
* Pui Ching Invitational Mathematics Competition (Official Website)
* Inter-school Mathematics Contest, a yearly contest organized by the Hong Kong Joint School Mathematics Society.
* Sing Yin Secondary School Invitational Mathematics Competition (Official Website)
[edit] Hungary
* Miklos Schweitzer Competition
* Nemzetközi Kenguru Matematika Verseny (3rd to 12th grade students)Homepage: http://www.zalamat.hu/
* Kalmár László Országos Matematika Verseny (3rd to 8th grade students)
* Zrínyi Ilona Országos Matematika Verseny (3rd to 8th grade students)
* Varga Tamás Matematika Verseny (7th and 8th grade students)
* Bátaszéki Matematika Verseny (3rd to 8th grade students)
* Középiskolai Matematikai Lapok (a year-long contest, each month you have to submit solutions to some problems, 9th-12th grade, Homepage in English: http://www.komal.hu/info/bemutatkozas.e.shtml )
* ADMV (Arany Dániel Matematika Verseny, 9th and 10th grade students)
* Gordiusz Matematika Tesztverseny (9th to 12th grade students)
* OKTV (Országos Középiskolai Tanulmányi Verseny, 11th and 12th grade)
* Kürschák József (first year university students or below)
[edit] India
[edit] Proof-based Mathematics Olympiads
* Regional Mathematical Olympiads held in every region. Leads to participation in Indian National Mathematical Olympiad, held every year as a part of selection process for International Mathematical Olympiad.[1]
* National Mathematics Talent Exams conducted by The Association of Mathematics Teachers of India, Chennai (For Vth to XIIth)
* National Genius Search Examination (NGSE) conducted by National Genius Search Foundation for Class V to X
* Gyanmania International Mathematics Olympiad (GIMO) conducted by Gyanwave.com for Age Group 15 to 23 Years.[2]
[edit] Application-based Mathematics Contests
* International Assessments for Indian Schools ( http://www.iais.emacmillan.com/mathematics_assessment.htm)
[edit] Indonesia
* Olimpiade Sains Nasional. A science olympiad held at the national level for wide variety of subjects, including mathematics. http://id.wikipedia.org/wiki/Olimpiade_Sains_Nasional
[edit] Iran
* Preliminary Mathematics Olympiad, which successful applicants compete against one another on Iranian Olympiad Level 2 competitions, then they proceeded to compete for top six places in the country so they could participate in the International Mathematics Olympiad as representatives of Iran. http://www.ysc.ac.ir
[edit] Ireland
* Irish Mathematical Societies Association Intervarsities. An annual event in which teams representing the mathematical societies of their respective colleges compete in an Olympiad styled event.
* The Irish Mathematical Olympiad (IrMO), an annual competition held in May each year (http://www.irmo.ie/). Students that perform well in the Junior Certificate are invited to take part in training programmes, leading up to the competition.
* Problem Solving for Irish Second Level Mathematicians (PRISM), a competition for secondary school students, organised by NUI Galway but held in the students' own schools. There are two competitions - one for junior students, and another for seniors (http://www.maths.nuigalway.ie/PRISM/).
* Team Math is held annually for secondary school students.
[edit] Israel
* The Grossman Olympiad for high school students.
* The Tournament of Towns (also called "Intercity Competitions in Math").
* The Gillis Math Olympiad for high-school students, organized by the Weizmann Institute of Science along with the Davidson Institute.
* The Zuta Math Olympiad for middle-school students, organized by the Weizmann Institute of Science along with the Davidson Institute.
* The Orange Math Olympiad, organized by the "Orange" Partner company of cellphones.
* The University Students Competition, supervised by the Israeli Mathematical Union.
* The Tel-Aviv University also participates in the William Lowell Putnam Mathematical Competition.
[edit] Italy
* Olimpiadi Italiane della Matematica (Italian Mathematical Olympiad). This is the national qualifying stage of the International Mathematical Olympiad.
* Campionati Internazionali di Giochi Matematici (International Mathematical Game Championship). This is the national qualifying stage of the Championnat International de Jeux Mathématiques et Logiques.
* Kangourou della Matematica (Mathematics Kangaroo). This is the national version of the Mathematical Kangaroo.
[edit] Japan
* JMO (Japanese Mathematical Olympiad)
* Kinki University Mathematics Contest
[edit] South Korea
* KMO (Korean Mathematical Olympiad, http://www.kms.or.kr/home/kmo/)
* KMC (Korean Mathematics Competition, http://www.kmath.co.kr/)
* KME (Korean Mathematics Evaluation, http://www.kerei.net/)
[edit] Macau, China
* Macao Mathematical Olympiad (Chinese homepage: http://www.sftw.umac.mo/~fstitl/olympiad/index.html )
[edit] Macedonia
* Regional Competition
* Republic Competition (different problems for each grade)
* JMMO (Junior Macedonian Maths Olympiad) (all students younger than 15.5 years have same questions)
* MMO (Macedonian Maths Olympiad) (all students older than 15.5 years have same questions)
Official web-site (in Macedonian): http://smm.org.mk/
[edit] Malaysia
* OMK (Olimpiad Matematik Kebangsaan / National Mathematical Olympiad), an annual competition organized by the Malaysian Mathematical Sciences Society, http://www.persama.org.my
* IMC (IIUM Mathematics Competition), organized by International Islamic University Malaysia, http://www.iiu.edu.my/imc
* MASMO (Malaysia ASEAN Schools Math Olympiads) http://www.masmo.info
* Hua Lo-Keng Cup Mathematics Competition (an annual competition organized by the Selangor-KL Hokkien Association)
* UTAR National Mathematics Competition (an annual competition organized by the University Tunku Abdul Rahman)
[edit] Mexico
* MMO (Mexican Mathematical Olympiad ( http://www.omm.unam.mx/ ) in Spanish
* Mathcounts - held annually at the American school of Puebla and open to ASOMEX students.
* ONMAS (Olimpiada nacional de matematicas para alumnos de secundaria)
* Canguro Matematico (Math Kangaro)
* Pierre Fermat Contest organized by IPN
* Olimpiada de Mayo(The selective exam for Olimpiada Rioplatense de Matematias)
* National Mathematical Tournament UAG
[edit] Myanmar
[edit] Netherlands
* NWO (Nederlandse Wiskunde Olympiade / Dutch Mathematics Olympiad, http://wiskundeolympiade.nl/)
* LIMO (Landelijke Interuniversitaire Mathematische Olympiade / National Interuniversitairy Mathematical Olympiad, http://www.limo.a-eskwadraat.nl/)
[edit] New Zealand
* Australasian Problem Solving Mathematical Olympiads (APSMO) (http://www.apsmo.info)
* Otago Problem Challenge (http://www.maths.otago.ac.nz/pc/)
* Singapore - Asia Pacific Mathematical Olympiad for Primary Schools (APMOPS: http://www.hci.sg/aphelion/apmops/)
* Auckland Mathematical Olympiad
* Eton Press Senior Mathematics Competition
* Australian Mathematics Competition (http://www.amt.canberra.edu.au/)
* National Bank Junior Mathematics Competition (NBJMC) (http://www.maths.otago.ac.nz/nbjmc/JMChome.php)
* September Problems (IMO Selection)
[edit] Norway
* Niels Henrik Abels matematikk-konkurranse (Norwegian Mathematical Olympiad, website available both in Norwegian and English at http://abelkonkurransen.no/)
[edit] Pakistan
* In 2006 almost 4,000,000 students from 41 countries played the game world-wide. The world "Kangaroo" center, which coordinates the competition in the various countries, was founded in 1994 in Paris. In Pakistan, the competition was first organized in 2005 by the Pakistan kangaroo Commission. For further information visit [3]
[edit] Paraguay
* Olimpiadas Matemáticas Paraguayas (OMAPA) [4]
[edit] Peru
* National Mathematic Olympiad - ONEM (Olimpiada Nacional Escort de Mathematical)
It's the official Olympiad, organized by the Ministry of Education and the Peruvian Mathematical Society in 4 phases. The final phase takes place near to Lima usually in November. Website available in Spanish: http://portal.huascaran.edu.pe/olimpiadas/index.htm hmm
[edit] Philippines
* Southern Tagalog Invitational Mathematical Challenge (Highscool Level)-UPLB Mathematical Sciences Society [www.uplbmass.org]
* Philippine Mathematical Olympiad
* Metrobank-Mathematics Teachers Association of the Philippines (MTAP)-Department of Education (DepEd) Math Challenge for Grade School and High School Students
* Annual Nationwide Search for the Math Wizard (College Level) - University of the Philippines Mathematics Club [5]
* mathematics festival
* MATHirang MATHibay - University of the Philippines Mathematics Majors' Circle
* MATHira MATHibay and STATstruck - Pamantasan ng Lungsod ng Maynila - Mathematical Society
* PUP MathMax
* Ateneo Mathematics Olympiad
[edit] Poland
* Polish Mathematical Olympiad (Website)
[edit] Portugal
* Portuguese Mathematics Olympiad - Olimpíadas Portuguesas da Matemática (in Portuguese)
* Olimpíada Paulistas (with Brazil)
[edit] Puerto Rico
* Puerto Rican Mathematical Olympiad - ompr.comoj.com (Olimpiadas Matemáticas de Puerto Rico (in Spanish))
[edit] Romania
* Romanian National Olympiad (http://www.gazetamatematica.net/?q=forum/1019)
* Annual Contest of Gazeta Matematica "Nicolae Teodorescu" (http://www.gazetamatematica.net/?q=node/28)
* The Mathematical Contest Traian Lalescu (http://www.colegiulnegruzzi.ro/index.php/negruzzi/activitati/premianti-subiecte-si-rezolvari/)
[edit] Russian Federation
* All-Russian Mathematical Olympiad (http://math.rusolymp.ru/)
* St.Petersburg Math Olympiad (http://www.pdmi.ras.ru/~olymp/index.html)
* Moscow Math Olympiad (http://olympiads.mccme.ru/mmo/)
* Leonard Euler Math Olympiad (http://www.matol.ru/)
[edit] Serbia
* Mathematical competition's in Serbia [6]
[edit] Singapore
* Singapore Mathematical Olympiad (SMO, http://sms.math.nus.edu.sg/Competitions/CompetitionHomePage.aspx)
* National Mathematical Olympiad of Singapore (NMOS, http://oas.highsch.nus.edu.sg/NMOS/)
[edit] Slovakia
* MO Matematická olympiáda http://matematika.okamzite.eu
* List of Slovak competitions http://www.cs.pitt.edu/~michal/projects/seminar/
* KMS Korešpondenčný matematický seminár http://kms.sk/
* STROM Korešpondenčný matematický seminár http://www.strom.sk
* MATMIX Korešpondenčný matematický seminár http://www.matmix.sk
* SEZAM Korešpondenčný matematický seminár http://www.sezam.sk
[edit] South Africa
* South African Mathematics Olympiad (SAMO, http://www.samf.ac.za/samo/)
* University of Cape Town Mathematics Competition (http://mth.uct.ac.za/competition/)
* Living Maths (http://www.livingmaths.com)
* Amesa (http://academic.sun.ac.za/mathed/AMESA/Challenge/Index.htm)
[edit] Sweden
* Skolornas matematiktävling (see http://www.math.uu.se/~dag/skolornas.html or http://www.math.chalmers.se/~sam/problemet/bok.html for older problems)
* Högstadiets matematiktävling (http://www.matematiktavling.org/hmt/, open for students in grades 7 through 9)
[edit] Taiwan
* Regional competition (http://umath.nuk.edu.tw/~senpengeu/HighSchool for some problems)
* National competition (http://umath.nuk.edu.tw/~senpengeu/HighSchool for some problems)
* IMO selection and training camp
[edit] Thailand
* 2008 International Mathematics Competition (IMC 2008) http://www.imcthai.net/
* National Mathematics Contest http://www.ipst.ac.th/olympic
* POSN-MO http://www.posn.or.th
* The Mathematical Association of Thailand Contest http://www.math.or.th/
* Diamond Crown http://www.obec.go.th/news48/05_may/27b/project_apply.pdf
* Sermpanya Mathematics Test http://www.sermpanya.com
* Mathcenter http://www.mathcenter.net/
[edit] Turkey
* Turkey National Mathematical Olympiad (Türkiye Ulusal Matematik Olimpiyatı (in Turkish), organized by TUBITAK) http://www.tubitak.gov.tr/bideb/
[edit] Ukraine
* Different Mathematics Competitions for school pupils http://matholymp.org.ua/
* Different Mathematics Competitions for university students http://putnam.ho.ua/
[edit] United Kingdom
* Most competitions are organised by the UK Mathematics Trust.
* The Primary Mathematics Challenge (for primary school pupils) is organised by the Mathematical Association.
* The Junior Mathematical Challenge is a multiple-choice competition for students up to year 8 in England and Wales, year S2 in Scotland, year 9 in Northern Ireland. High scorers in the JMC are invited to compete in the Junior Mathematical Olympiad.
* The Intermediate Mathematical Challenge is a multiple-choice competition for students up to year 11 in England and Wales, year S4 in Scotland, year 12 in Northern Ireland. High scorers in the IMC are invited to compete in the Intermediate Mathematical Olympiad and Kangaroo (for the highest scorers) and in the European Kangaroo (another multiple-choice competition, for other high scorers).
* The Senior Mathematical Challenge (formerly National Maths Contest) is a multiple-choice competition for students up to year 13 in England and Wales, year S6 in Scotland, year 14 in Northern Ireland.
* High scorers in the SMC are invited to compete in the British Mathematical Olympiad (http://www.bmoc.maths.org/).
* There is a Team Maths Challenge for students in England, Wales and Northern Ireland; Scotland has its own Enterprising Mathematics Competition organised by the Scottish Mathematical Council.
* The UCL Maths Challenge is a competition for Year 6 primary school pupils from London organised by UCL student volunteers (http://www.smnd.sk/kotanyi/maths_challenge/).
* Olimpiada Kombëtare e Matematikës dhe e Formimit të Përgjithshëm.
* Olimpiada Kombëtare e Matematikës
* Konkursi i Pranverës 1(BMO TST)
* Konkursi i Pranverës 2(IMO TST)
[edit] Argentina
* OMA (Olimpíada Matemática Argentina, http://www.oma.org.ar)
* Olimpiada Matematica Ñandu
* Certamen "El numero de oro"
* Torneo Computacion y Matematicas
[edit] Australia
* APSMO (Australasian Problem Solving Mathematical Olympiads, http://www.apsmo.info)
* AMC (Australian Mathematics Competition, http://www.amt.edu.au)
* AMO (Australian Mathematical Olympiad, http://www.amt.edu.au)
* AIMO (Australian Intermediate Mathematical Olympiad)
* UNSW School Mathematics Competition (http://www.maths.unsw.edu.au/highschool/unsw/highcomps.html)
* HSFOL (Have Sum Fun Online) - an online team competition from the Mathematics Association of West Australia (http://www.havesumfunonline.com/)
[edit] Austria
* ÖMO (Österreichische Mathematik Olympiade) (http://www.oemo.at)
[edit] Belgium
French-speaking students can compete in the OMB (Olympiade Mathématique Belge) consisting of three categories:
* Mini (grades 7 and 8)
* Mide (grades 9 and 10)
* Maxi (grades 11 and 12)
Dutch-speaking students can compete in the VWO (Vlaamse Wiskunde Olympiade), with two categories:
* Junior Wiskunde Olympiade (grades 9 and 10)
* Vlaamse Wiskunde Olympiade (grades 11 and 12)
[edit] Brazil
There are two national competitions in Brazil: the oldest one, OBM, dates back to 1979 and is open to all students, from fifth grade to university.
The other one, OBMEP, was created in 2005 and is open to public school students from fifth grade to high school. In 2008 counted with the participation of 18,3 million students on the first round.
* OBM (Olimpíada Brasileira de Matemática, http://www.obm.org.br/)
* OBMEP (Olimpíada Brasileira de Matemática das Escolas Públicas, http://www.obmep.org.br/)
There are also many regional competitions, usually open to all students of a given state.
* OPM (Olimpíada Paulista de Matemática, http://www.opm.mat.br/)
* OMERJ (Olimpíada de Matemática do Estado do Rio de Janeiro, http://www.omerj.com.br/)
[edit] Bulgaria
* Bulgarian Competition in Mathematics and Informatics (http://www.math.bas.bg/bcmi/)
[edit] Canada
National competitions hosted by Mathematica - The Mathematics Contest Centre:
Multiple Choice:
* Newton Contest (9th grade students)
* Lagrange Contest (8th grade students)
* Euler Contest (7th grade students)
* Pythagoras Contest (6th grade students)
* Fibonacci Contest (5th grade students)
* Byron-Germain Contest (4th grade students)
* Thales Contest (3rd grade students)
National competitions hosted by The Centre for Education in Mathematics and Computing:
Full Solutions:
* Euclid (12th grade students)
* Hypatia (11th grade students)
* Galois (10th grade students)
* Fryer (9th grade students)
Multiple Choice:
* Fermat (11th grade students)
* Cayley (10th grade students)
* Pascal (9th grade students)
* Gauss (7th and 8th grade students)
Other competitions hosted by The Canadian Mathematical Society:
* Canadian Open Mathematics Challenge
* Canadian Mathematics Olympiad, the official mathematical Olympiad of Canada and its International Mathematical Olympiad TST, or team selection test.
MATHChallengers (formerly MathCounts BC) is called MathChallengers since 2005. It is hosted by APEGBC.
[edit] China
* CMO (China Mathematics Olympiad 中国数学奥林匹克)
* CWMO (Western China Mathematics Olympiad)
* CGMO (China Girl Mathematics Olympiad) — for female secondary students
* CSMO (South-eastern China Mathematics Olympiad) — for secondary 4 students
* CNMO (China Northern Mathematics Olympiad)
* National high school math League 全国高中数学联赛
* QAIS (Qingdao American International School 青岛美国国际学校) stock market trading, bridge building and siege engines Grades 7 - 12
[edit] Colombia
* OCM (Colombian Mathematics Olympiad)
* OCMU (Colombian Mathematics Olympiad university)
Web site: http://olimpia.uan.edu.co/olimpiadas/public/frameset.jsp
[edit] Cyprus
* Cyprus Mathematical Olympiad — Mathematics competitions for students in every grade.
* Junior high-school (Gymnasium) Competitions — For students under 15.5 years old (1st, 2nd, and 3rd grade of Gymnasium)
* High School (Lyceum) Competitions — For students above 15.5 years old (1st, 2nd, and 3rd grade of Lyceum)
* European Kangourou Mathematics Competition
[edit] Czech Republic
* MO (Czech National Mathematics Olympiad, http://www.math.muni.cz/~rvmo)
* PraSe (Prague Mathematics Correspondence Seminar for Secondary School Students, http://mks.mff.cuni.cz)
* Brkos (Brno Mathematical Correspondence Seminar for Secondary School Students, http://www.math.muni.cz/~brkos)
[edit] Denmark
* Georg Mohr (Contest for Danish high school students, http://www.georgmohr.dk)
[edit] Germany
* DeMO (Deutsche Mathematik-Olympiade / German Math Olympiad, http://www.mathematik-olympiaden.de)
* BWM (Bundeswettbewerb Mathematik / Federal Math Competition , http://www.bundeswettbewerb-mathematik.de),
* AIMO (Auswahlwettbewerb zur Internationalen Mathematik-Olympiade / Team selection tests for the IMO, http://www.bundeswettbewerb-mathematik.de/imo/main.htm)
* LWMB (Landeswettbewerb Mathematik Bayern / Bavarian Math Competition, http://lwmb.de), LWM (Landeswettbewerb Mathematik Baden-Württemberg / Math Competition of Baden-Württemberg, http://landeswettbewerb-mathematik.de)
[edit] Greece
* Θαλής (Thales) - first round
* Ευκλείδης (Euclides) - second round
* Αρχιμήδης (Archimedes) - third round
* Λευκοπούλειος Διαγωνισμός Πιθανοτήτων και Στατιστικής - "Leukopouleios" Contest on Probabilities And Statistics - a competition that is not related to IMO, organised by ESI (National Statistic Institut)
* Math Kangaroo Competition
Seen also Hellenic Mathematical Society
[edit] Hong Kong
* Hong Kong Mathematics Olympiad
* The Hong Kong Mathematical High Achievers Selection Contest. Only Form 1 to form 3 students may participate in this contest.
* International Mathematical Olympiad Preliminary Selection Contest - Hong Kong (IMO HK Prelim, Official Website)
* Pui Ching Invitational Mathematics Competition (Official Website)
* Inter-school Mathematics Contest, a yearly contest organized by the Hong Kong Joint School Mathematics Society.
* Sing Yin Secondary School Invitational Mathematics Competition (Official Website)
[edit] Hungary
* Miklos Schweitzer Competition
* Nemzetközi Kenguru Matematika Verseny (3rd to 12th grade students)Homepage: http://www.zalamat.hu/
* Kalmár László Országos Matematika Verseny (3rd to 8th grade students)
* Zrínyi Ilona Országos Matematika Verseny (3rd to 8th grade students)
* Varga Tamás Matematika Verseny (7th and 8th grade students)
* Bátaszéki Matematika Verseny (3rd to 8th grade students)
* Középiskolai Matematikai Lapok (a year-long contest, each month you have to submit solutions to some problems, 9th-12th grade, Homepage in English: http://www.komal.hu/info/bemutatkozas.e.shtml )
* ADMV (Arany Dániel Matematika Verseny, 9th and 10th grade students)
* Gordiusz Matematika Tesztverseny (9th to 12th grade students)
* OKTV (Országos Középiskolai Tanulmányi Verseny, 11th and 12th grade)
* Kürschák József (first year university students or below)
[edit] India
[edit] Proof-based Mathematics Olympiads
* Regional Mathematical Olympiads held in every region. Leads to participation in Indian National Mathematical Olympiad, held every year as a part of selection process for International Mathematical Olympiad.[1]
* National Mathematics Talent Exams conducted by The Association of Mathematics Teachers of India, Chennai (For Vth to XIIth)
* National Genius Search Examination (NGSE) conducted by National Genius Search Foundation for Class V to X
* Gyanmania International Mathematics Olympiad (GIMO) conducted by Gyanwave.com for Age Group 15 to 23 Years.[2]
[edit] Application-based Mathematics Contests
* International Assessments for Indian Schools ( http://www.iais.emacmillan.com/mathematics_assessment.htm)
[edit] Indonesia
* Olimpiade Sains Nasional. A science olympiad held at the national level for wide variety of subjects, including mathematics. http://id.wikipedia.org/wiki/Olimpiade_Sains_Nasional
[edit] Iran
* Preliminary Mathematics Olympiad, which successful applicants compete against one another on Iranian Olympiad Level 2 competitions, then they proceeded to compete for top six places in the country so they could participate in the International Mathematics Olympiad as representatives of Iran. http://www.ysc.ac.ir
[edit] Ireland
* Irish Mathematical Societies Association Intervarsities. An annual event in which teams representing the mathematical societies of their respective colleges compete in an Olympiad styled event.
* The Irish Mathematical Olympiad (IrMO), an annual competition held in May each year (http://www.irmo.ie/). Students that perform well in the Junior Certificate are invited to take part in training programmes, leading up to the competition.
* Problem Solving for Irish Second Level Mathematicians (PRISM), a competition for secondary school students, organised by NUI Galway but held in the students' own schools. There are two competitions - one for junior students, and another for seniors (http://www.maths.nuigalway.ie/PRISM/).
* Team Math is held annually for secondary school students.
[edit] Israel
* The Grossman Olympiad for high school students.
* The Tournament of Towns (also called "Intercity Competitions in Math").
* The Gillis Math Olympiad for high-school students, organized by the Weizmann Institute of Science along with the Davidson Institute.
* The Zuta Math Olympiad for middle-school students, organized by the Weizmann Institute of Science along with the Davidson Institute.
* The Orange Math Olympiad, organized by the "Orange" Partner company of cellphones.
* The University Students Competition, supervised by the Israeli Mathematical Union.
* The Tel-Aviv University also participates in the William Lowell Putnam Mathematical Competition.
[edit] Italy
* Olimpiadi Italiane della Matematica (Italian Mathematical Olympiad). This is the national qualifying stage of the International Mathematical Olympiad.
* Campionati Internazionali di Giochi Matematici (International Mathematical Game Championship). This is the national qualifying stage of the Championnat International de Jeux Mathématiques et Logiques.
* Kangourou della Matematica (Mathematics Kangaroo). This is the national version of the Mathematical Kangaroo.
[edit] Japan
* JMO (Japanese Mathematical Olympiad)
* Kinki University Mathematics Contest
[edit] South Korea
* KMO (Korean Mathematical Olympiad, http://www.kms.or.kr/home/kmo/)
* KMC (Korean Mathematics Competition, http://www.kmath.co.kr/)
* KME (Korean Mathematics Evaluation, http://www.kerei.net/)
[edit] Macau, China
* Macao Mathematical Olympiad (Chinese homepage: http://www.sftw.umac.mo/~fstitl/olympiad/index.html )
[edit] Macedonia
* Regional Competition
* Republic Competition (different problems for each grade)
* JMMO (Junior Macedonian Maths Olympiad) (all students younger than 15.5 years have same questions)
* MMO (Macedonian Maths Olympiad) (all students older than 15.5 years have same questions)
Official web-site (in Macedonian): http://smm.org.mk/
[edit] Malaysia
* OMK (Olimpiad Matematik Kebangsaan / National Mathematical Olympiad), an annual competition organized by the Malaysian Mathematical Sciences Society, http://www.persama.org.my
* IMC (IIUM Mathematics Competition), organized by International Islamic University Malaysia, http://www.iiu.edu.my/imc
* MASMO (Malaysia ASEAN Schools Math Olympiads) http://www.masmo.info
* Hua Lo-Keng Cup Mathematics Competition (an annual competition organized by the Selangor-KL Hokkien Association)
* UTAR National Mathematics Competition (an annual competition organized by the University Tunku Abdul Rahman)
[edit] Mexico
* MMO (Mexican Mathematical Olympiad ( http://www.omm.unam.mx/ ) in Spanish
* Mathcounts - held annually at the American school of Puebla and open to ASOMEX students.
* ONMAS (Olimpiada nacional de matematicas para alumnos de secundaria)
* Canguro Matematico (Math Kangaro)
* Pierre Fermat Contest organized by IPN
* Olimpiada de Mayo(The selective exam for Olimpiada Rioplatense de Matematias)
* National Mathematical Tournament UAG
[edit] Myanmar
[edit] Netherlands
* NWO (Nederlandse Wiskunde Olympiade / Dutch Mathematics Olympiad, http://wiskundeolympiade.nl/)
* LIMO (Landelijke Interuniversitaire Mathematische Olympiade / National Interuniversitairy Mathematical Olympiad, http://www.limo.a-eskwadraat.nl/)
[edit] New Zealand
* Australasian Problem Solving Mathematical Olympiads (APSMO) (http://www.apsmo.info)
* Otago Problem Challenge (http://www.maths.otago.ac.nz/pc/)
* Singapore - Asia Pacific Mathematical Olympiad for Primary Schools (APMOPS: http://www.hci.sg/aphelion/apmops/)
* Auckland Mathematical Olympiad
* Eton Press Senior Mathematics Competition
* Australian Mathematics Competition (http://www.amt.canberra.edu.au/)
* National Bank Junior Mathematics Competition (NBJMC) (http://www.maths.otago.ac.nz/nbjmc/JMChome.php)
* September Problems (IMO Selection)
[edit] Norway
* Niels Henrik Abels matematikk-konkurranse (Norwegian Mathematical Olympiad, website available both in Norwegian and English at http://abelkonkurransen.no/)
[edit] Pakistan
* In 2006 almost 4,000,000 students from 41 countries played the game world-wide. The world "Kangaroo" center, which coordinates the competition in the various countries, was founded in 1994 in Paris. In Pakistan, the competition was first organized in 2005 by the Pakistan kangaroo Commission. For further information visit [3]
[edit] Paraguay
* Olimpiadas Matemáticas Paraguayas (OMAPA) [4]
[edit] Peru
* National Mathematic Olympiad - ONEM (Olimpiada Nacional Escort de Mathematical)
It's the official Olympiad, organized by the Ministry of Education and the Peruvian Mathematical Society in 4 phases. The final phase takes place near to Lima usually in November. Website available in Spanish: http://portal.huascaran.edu.pe/olimpiadas/index.htm hmm
[edit] Philippines
* Southern Tagalog Invitational Mathematical Challenge (Highscool Level)-UPLB Mathematical Sciences Society [www.uplbmass.org]
* Philippine Mathematical Olympiad
* Metrobank-Mathematics Teachers Association of the Philippines (MTAP)-Department of Education (DepEd) Math Challenge for Grade School and High School Students
* Annual Nationwide Search for the Math Wizard (College Level) - University of the Philippines Mathematics Club [5]
* mathematics festival
* MATHirang MATHibay - University of the Philippines Mathematics Majors' Circle
* MATHira MATHibay and STATstruck - Pamantasan ng Lungsod ng Maynila - Mathematical Society
* PUP MathMax
* Ateneo Mathematics Olympiad
[edit] Poland
* Polish Mathematical Olympiad (Website)
[edit] Portugal
* Portuguese Mathematics Olympiad - Olimpíadas Portuguesas da Matemática (in Portuguese)
* Olimpíada Paulistas (with Brazil)
[edit] Puerto Rico
* Puerto Rican Mathematical Olympiad - ompr.comoj.com (Olimpiadas Matemáticas de Puerto Rico (in Spanish))
[edit] Romania
* Romanian National Olympiad (http://www.gazetamatematica.net/?q=forum/1019)
* Annual Contest of Gazeta Matematica "Nicolae Teodorescu" (http://www.gazetamatematica.net/?q=node/28)
* The Mathematical Contest Traian Lalescu (http://www.colegiulnegruzzi.ro/index.php/negruzzi/activitati/premianti-subiecte-si-rezolvari/)
[edit] Russian Federation
* All-Russian Mathematical Olympiad (http://math.rusolymp.ru/)
* St.Petersburg Math Olympiad (http://www.pdmi.ras.ru/~olymp/index.html)
* Moscow Math Olympiad (http://olympiads.mccme.ru/mmo/)
* Leonard Euler Math Olympiad (http://www.matol.ru/)
[edit] Serbia
* Mathematical competition's in Serbia [6]
[edit] Singapore
* Singapore Mathematical Olympiad (SMO, http://sms.math.nus.edu.sg/Competitions/CompetitionHomePage.aspx)
* National Mathematical Olympiad of Singapore (NMOS, http://oas.highsch.nus.edu.sg/NMOS/)
[edit] Slovakia
* MO Matematická olympiáda http://matematika.okamzite.eu
* List of Slovak competitions http://www.cs.pitt.edu/~michal/projects/seminar/
* KMS Korešpondenčný matematický seminár http://kms.sk/
* STROM Korešpondenčný matematický seminár http://www.strom.sk
* MATMIX Korešpondenčný matematický seminár http://www.matmix.sk
* SEZAM Korešpondenčný matematický seminár http://www.sezam.sk
[edit] South Africa
* South African Mathematics Olympiad (SAMO, http://www.samf.ac.za/samo/)
* University of Cape Town Mathematics Competition (http://mth.uct.ac.za/competition/)
* Living Maths (http://www.livingmaths.com)
* Amesa (http://academic.sun.ac.za/mathed/AMESA/Challenge/Index.htm)
[edit] Sweden
* Skolornas matematiktävling (see http://www.math.uu.se/~dag/skolornas.html or http://www.math.chalmers.se/~sam/problemet/bok.html for older problems)
* Högstadiets matematiktävling (http://www.matematiktavling.org/hmt/, open for students in grades 7 through 9)
[edit] Taiwan
* Regional competition (http://umath.nuk.edu.tw/~senpengeu/HighSchool for some problems)
* National competition (http://umath.nuk.edu.tw/~senpengeu/HighSchool for some problems)
* IMO selection and training camp
[edit] Thailand
* 2008 International Mathematics Competition (IMC 2008) http://www.imcthai.net/
* National Mathematics Contest http://www.ipst.ac.th/olympic
* POSN-MO http://www.posn.or.th
* The Mathematical Association of Thailand Contest http://www.math.or.th/
* Diamond Crown http://www.obec.go.th/news48/05_may/27b/project_apply.pdf
* Sermpanya Mathematics Test http://www.sermpanya.com
* Mathcenter http://www.mathcenter.net/
[edit] Turkey
* Turkey National Mathematical Olympiad (Türkiye Ulusal Matematik Olimpiyatı (in Turkish), organized by TUBITAK) http://www.tubitak.gov.tr/bideb/
[edit] Ukraine
* Different Mathematics Competitions for school pupils http://matholymp.org.ua/
* Different Mathematics Competitions for university students http://putnam.ho.ua/
[edit] United Kingdom
* Most competitions are organised by the UK Mathematics Trust.
* The Primary Mathematics Challenge (for primary school pupils) is organised by the Mathematical Association.
* The Junior Mathematical Challenge is a multiple-choice competition for students up to year 8 in England and Wales, year S2 in Scotland, year 9 in Northern Ireland. High scorers in the JMC are invited to compete in the Junior Mathematical Olympiad.
* The Intermediate Mathematical Challenge is a multiple-choice competition for students up to year 11 in England and Wales, year S4 in Scotland, year 12 in Northern Ireland. High scorers in the IMC are invited to compete in the Intermediate Mathematical Olympiad and Kangaroo (for the highest scorers) and in the European Kangaroo (another multiple-choice competition, for other high scorers).
* The Senior Mathematical Challenge (formerly National Maths Contest) is a multiple-choice competition for students up to year 13 in England and Wales, year S6 in Scotland, year 14 in Northern Ireland.
* High scorers in the SMC are invited to compete in the British Mathematical Olympiad (http://www.bmoc.maths.org/).
* There is a Team Maths Challenge for students in England, Wales and Northern Ireland; Scotland has its own Enterprising Mathematics Competition organised by the Scottish Mathematical Council.
* The UCL Maths Challenge is a competition for Year 6 primary school pupils from London organised by UCL student volunteers (http://www.smnd.sk/kotanyi/maths_challenge/).
Regional mathematics competitions
* AITMO (Asian Inter-city Teenagers Mathematics Olympiad) — for junior secondary students around the Eastern Asian region
* APMO (Asian Pacific Mathematics Olympiad) — Pacific rim
* Balkan Mathematical Olympiad — for students from 15.5 years old from Balkan area
* Baltic Way — Baltic area
* ICAS-Mathematics (http://www.eaa.unsw.edu.au/about_icas/mathematics, formerly Australasian Schools Mathematics Assessment)
* Junior Balkan Mathematical Olympiad — for students under 15.5 years old from Balkan area
* MEMO (Middle European Mathematical Olympiad) — Germany, Croatia, Austria, Poland, Switzerland, Slovakia, Slovenia, Czech Republic, Hungary
* NMC (Nordic Mathematical Contest) — the five Nordic countries
* Nordic university-level mathematics team-competition — For Nordic undergraduates (http://cc.oulu.fi/~phasto/competition/)
* OIM (Olimpíadas Iberoamericanas de Matemática) — Spain, Portugal and Latin America (http://www.oei.es/oim/index.html)
* Olimpiada de mayo (competitions for selecting the participants of Olimpiada Matematica Rioplatence) (http://www.oma.org.ar/internacional/may.htm)
* Olimpiada Iberoamericana de Matematicas para Estudiantes Universitarios (similar to Olimpiada Iberoamericana de Matematica, but it is for collage students)
* Olimpiada Matematica Rioplatense (similar to Olimpiada Iberoamericana de Matematica, but it is every year in Argentina and participants are organized in levels depending on the age) (http://www.oma.org.ar/internacional/omr.htm)
* Olimpiada Matematica de Centroamérica y del Caribe — Central America and the Caribbean
* Olimpiada Matematica de Paises del Cono Sur — 8 countries from South America
* SEAMO (SEAMEO Mathematics Olympiad) — South-East Asia
* SEAMC (South East Asian Mathematics Competition) — South-East Asia (http://hseagle.sas.edu.sg/seamc/)
* William Lowell Putnam Mathematical Competition — USA and Canada (http://www.maa.org/awards/putnam.html)
* APMOPS (Singapore - Asia Pacific Mathematical Olympiad for Primary Schools) — open to Primary Students under 12 in Australia, Brunei, China (Shanghai, Hainan, Xiamen, Wenchou), Hong Kong, Indonesia (Jakarta), Malaysia: (Johor, Kuala Lumpur, Selangor- Petaling Jaya), Penang, Perak- Ipoh, Kedah), New Zealand, Singapore, South Korea, Taiwan, The Philippines & India. (http://www.hci.sg/aphelion/apmops/index.htm)
* ZIMO (Zhautykov International Mathematical Olympiad) — for teams from specialized schools in post-soviet region (http://izho2008.fiz-mat.kz/)
* Hungary-Israel Mathematical Competition. It was established in 1990. Only this 2 countries participate and one is the host. It is hold in spring. It consist of individual and team contests.
* Tuymaada Yakut Olympiad. Multidisciplinary competition for students from Romania, Kazakhstan, Moldova, Tartastan, Saint Petersburg, Irkutsk, Vladivostok, Novokuznetsk, Perm and other Russian cities. It is held in July; few students obtain prizes.
* Donova Mathematical Olympiad. Olympiad for all countries through which the Danube passes. Held since 2005.
* Mediterranean Mathematics Olympiad. Olympiad for countries in the Mediterranean zone.
* PAMO (Pan African Mathematics Olympiad)
* APMC (Austrian-Polish Mathematics Competition) (last held in 2006)
* Czech-Polish-Slovak Match. Established in 1995 under the name of Czech-Slovak Match Poland joined in 2001. It is held in June in the IMO format.
* APMO (Asian Pacific Mathematics Olympiad) — Pacific rim
* Balkan Mathematical Olympiad — for students from 15.5 years old from Balkan area
* Baltic Way — Baltic area
* ICAS-Mathematics (http://www.eaa.unsw.edu.au/about_icas/mathematics, formerly Australasian Schools Mathematics Assessment)
* Junior Balkan Mathematical Olympiad — for students under 15.5 years old from Balkan area
* MEMO (Middle European Mathematical Olympiad) — Germany, Croatia, Austria, Poland, Switzerland, Slovakia, Slovenia, Czech Republic, Hungary
* NMC (Nordic Mathematical Contest) — the five Nordic countries
* Nordic university-level mathematics team-competition — For Nordic undergraduates (http://cc.oulu.fi/~phasto/competition/)
* OIM (Olimpíadas Iberoamericanas de Matemática) — Spain, Portugal and Latin America (http://www.oei.es/oim/index.html)
* Olimpiada de mayo (competitions for selecting the participants of Olimpiada Matematica Rioplatence) (http://www.oma.org.ar/internacional/may.htm)
* Olimpiada Iberoamericana de Matematicas para Estudiantes Universitarios (similar to Olimpiada Iberoamericana de Matematica, but it is for collage students)
* Olimpiada Matematica Rioplatense (similar to Olimpiada Iberoamericana de Matematica, but it is every year in Argentina and participants are organized in levels depending on the age) (http://www.oma.org.ar/internacional/omr.htm)
* Olimpiada Matematica de Centroamérica y del Caribe — Central America and the Caribbean
* Olimpiada Matematica de Paises del Cono Sur — 8 countries from South America
* SEAMO (SEAMEO Mathematics Olympiad) — South-East Asia
* SEAMC (South East Asian Mathematics Competition) — South-East Asia (http://hseagle.sas.edu.sg/seamc/)
* William Lowell Putnam Mathematical Competition — USA and Canada (http://www.maa.org/awards/putnam.html)
* APMOPS (Singapore - Asia Pacific Mathematical Olympiad for Primary Schools) — open to Primary Students under 12 in Australia, Brunei, China (Shanghai, Hainan, Xiamen, Wenchou), Hong Kong, Indonesia (Jakarta), Malaysia: (Johor, Kuala Lumpur, Selangor- Petaling Jaya), Penang, Perak- Ipoh, Kedah), New Zealand, Singapore, South Korea, Taiwan, The Philippines & India. (http://www.hci.sg/aphelion/apmops/index.htm)
* ZIMO (Zhautykov International Mathematical Olympiad) — for teams from specialized schools in post-soviet region (http://izho2008.fiz-mat.kz/)
* Hungary-Israel Mathematical Competition. It was established in 1990. Only this 2 countries participate and one is the host. It is hold in spring. It consist of individual and team contests.
* Tuymaada Yakut Olympiad. Multidisciplinary competition for students from Romania, Kazakhstan, Moldova, Tartastan, Saint Petersburg, Irkutsk, Vladivostok, Novokuznetsk, Perm and other Russian cities. It is held in July; few students obtain prizes.
* Donova Mathematical Olympiad. Olympiad for all countries through which the Danube passes. Held since 2005.
* Mediterranean Mathematics Olympiad. Olympiad for countries in the Mediterranean zone.
* PAMO (Pan African Mathematics Olympiad)
* APMC (Austrian-Polish Mathematics Competition) (last held in 2006)
* Czech-Polish-Slovak Match. Established in 1995 under the name of Czech-Slovak Match Poland joined in 2001. It is held in June in the IMO format.
International mathematics competitions
- AIMMS/MOPTA Optimization Modelling Competition for university students, held annually. (http://mopta.ie.lehigh.edu)
- Vojtěch Jarník International Mathematical Competition — international competition for undergraduate students. The Competition is held at the University of Ostrava every year in March or April. (http://vjimc.osu.cz/)
- China Girls Math Olympiad (CGMO) — Olympiad held annually in different cities in China for teams of girls representing regions within China and a number of other countries as well.
- World Mathematics Challenge (WMC) — an international competition for high school students.
- International Mathematics Competition for University Students (IMC) — international competition for undergraduate students. (http://www.imc-math.org.uk/)
- International Scientific Olympiad on Mathematics for Undergraduate University Students (ISOM) — competition for undergraduates held annually in Iran. (http://olympiad.sanjesh.org/en/index.asp)
- International Schools Mathematics Teachers Foundation (ISMTF) — annual competition for high school students attending an international school. Held every year in a different school. (http://www.ismtf.org/index.html)
- Invitational World Youth Mathematics Intercity Competition (IWYMIC) — held annually in different cities in China for students under 15.5 years old worldwide. (http://www.iwymicsa.co.za/)
- South Eastern European Mathematical Olympiad for First and Second Year University Students with International Participation (SEEMOUS) — competition for the Balkan region; however, participation is international. The first Olympiad was held in Agros, Cyprus, 7–12 March 2007, the second in Athens, Greece, 5–10 March 2008, the third in Agros, Cyprus, 4–9 March 2009, the fourth in Plovdiv, Bulgaria, 8–13 March 2010, the fifth is to be held in Bucharest, Romania in 2011. (http://www.seemous.eu/ and http://seemous2010.fmi-plovdiv.org/)
- International Mathematical Olympiad (IMO) — The oldest international Olympiad, occurring annually since 1959.
- Mathematical Contest in Modeling (MCM) — team contest for undergraduates (http://www.comap.com/undergraduate/contests/mcm/)
- Interdisciplinary Contest in Modeling (ICM) — team contest for undergraduates (http://www.comap.com/undergraduate/contests/icm/)
- Purple Comet! Math Meet — annual on-line team contest for high school & middle school (http://purplecomet.org/)
- Math exercises for kids — Daily worldwide competition (http://www.math-exercises-for-kids.com/competition/math-world-cup.php)
- Primary Mathematics World Contest (PMWC) — worldwide competition.
- Tournament of the Towns — worldwide competition.
- Mathematical Kangaroo — worldwide competition.
- Championnat International de Jeux Mathématiques et Logiques — for all ages, mainly for French-speaking countries, but participation is not limited by language. (http://ffjm.cijm.org)
- Romanian Master of Sciences — This is an olympiad for the selections of the 20 top countries in the last IMO. The level of the competition is IMO-like. The format had been 4 problems in 5 hours in 2009, in 2010 it was changed to 3 problems in 4 hours, two days format.(http://www.rmm.lbi.ro/)
- Rocket City Math League (RCML) — This is a mathematics competition run by students at Virgil I. Grissom High School with levels ranging from Explorer (Pre-Algebra) to Discovery (Comprehensive). (http://www.rocketcitymath.org)
- Mental Calculation World Cup — a contest for the best mental calculators
Mathematics Olympiad
The Mathematics Olympiad activity was undertaken by NBHM from 1986 onwards and is currently run in collaboration with the Homi Bhabha Centre for Science Education, Mumbai. One main purpose of this activity is to support mathematical talent among high school students in the country. NBHM has taken on the responsibility for selecting and training the Indian team for participation in the International Mathematical Olympiad every year. While NBHM coordinates and supports Mathematics Olympiad contests all over the country, regional bodies, mostly voluntary, play an important role at different stages. For the purpose of administering Mathematics Olympiad contests, the country has been divided in 16 regions. A regional coordinator is responsible for conducting these tests in each region. The names of the regions and their respective regional coordinators are given at the end.General Information about Mathematics ContestsThe Mathematics Olympiad Programme leading to participation in the International Mathematical Olympiad consists of the following stages:
Stage 1: Regional Mathematical Olympiad (RMO)RMO is held in each region normally between September and the first Sunday of December each year. The regional coordinator ensures that at least one centre is provided in each district of the region. All high school students up to class XII are eligible to appear for RMO. RMO is a 3-hour written test containing about 6 to 7 problems. Each regional coordinator has the freedom to prepare his/her own question paper or to obtain the question paper from NBHM. The regions opting for the NBHM question paper hold this contest on the 1st Sunday of December. On the basis of the performance in RMO, a certain number of students from each region are selected to appear for the second stage. Regional coordinators charge nominal fees to meet the expenses for organizing the contests.
Stage 2: Indian National Mathematical Olympiad (INMO)INMO is held on the first Sunday of February each year at various Centres in different regions. Only students selected on the basis of RMO from different regions are eligible to appear for INMO. INMO is a 4-hour written test. The question paper is set centrally and is common throughout the country. The top 30-35 performers in INMO receive a certificate of merit.
Stage 3: International Mathematical Olympiad Training Camp (IMOTC)The INMO certificate awardees are invited to a month long training camp (junior batch) conducted in May-June, each year. In addition, INMO awardees of the previous year that have satisfactorily gone through postal tuition throughout the year are invited again for a second round of training (senior batch).
Stage 4: International Mathematical Olympiad (IMO)The team selected at the end of the camp, a "leader" and a "deputy leader," represent India at the IMO that is normally held in July in a different member country of IMO each year. The leader and deputy leader are chosen by NBHM from among mathematics teachers/researchers involved in the Mathematics Olympiad activity. IMO consists of two written tests held on two days with a gap of at least one day. Each test is of four-and-a-half-hours duration. Travel to IMO venue and return takes about two weeks. India has been participating in IMO since 1989. Students of the Indian team who receive gold, silver and bronze medals at IMO receive a cash prize of Rs. 5,000/-, Rs. 4,000/- and Rs. 3,000/- respectively, from NBHM during the following year at a formal ceremony at the end of the training camp.The Ministry of Human Resource Development (MHRD) finances international travel of the eight-member Indian delegation connected with international participation. NBHM finances the entire in-country programme and takes care of other expenditureStudents aiming for selection for participation in IMO should note that RMO is the first essential step for the programme. To appear for RMO, students should get in touch with the RMO co-ordinator of their region well in advance, for enrolment and payment of a nominal fee.Syllabus for Mathematics OlympiadsThe syllabus for Mathematics Olympiads (regional, national and international) is pre-degree college mathematics. The areas covered are: number systems, arithmetic of integers, geometry, quadratic equations and expressions, trigonometry, co-ordinate geometry, systems of linear equations, permutations and combinations, factorisation of polynomials, inequalities, elementary combinatorics, probability theory, number theory, infinite series, complex numbers and elementary graph theory. The syllabus does not include calculus and statistics. The typical areas for problems are: number theory, geometry, algebra and combinatorics. The syllabus is in a sense spread over class IX to class XII levels, but the problems under each topic are of an exceptionally high level in difficulty and sophistication.
Stage 1: Regional Mathematical Olympiad (RMO)RMO is held in each region normally between September and the first Sunday of December each year. The regional coordinator ensures that at least one centre is provided in each district of the region. All high school students up to class XII are eligible to appear for RMO. RMO is a 3-hour written test containing about 6 to 7 problems. Each regional coordinator has the freedom to prepare his/her own question paper or to obtain the question paper from NBHM. The regions opting for the NBHM question paper hold this contest on the 1st Sunday of December. On the basis of the performance in RMO, a certain number of students from each region are selected to appear for the second stage. Regional coordinators charge nominal fees to meet the expenses for organizing the contests.
Stage 2: Indian National Mathematical Olympiad (INMO)INMO is held on the first Sunday of February each year at various Centres in different regions. Only students selected on the basis of RMO from different regions are eligible to appear for INMO. INMO is a 4-hour written test. The question paper is set centrally and is common throughout the country. The top 30-35 performers in INMO receive a certificate of merit.
Stage 3: International Mathematical Olympiad Training Camp (IMOTC)The INMO certificate awardees are invited to a month long training camp (junior batch) conducted in May-June, each year. In addition, INMO awardees of the previous year that have satisfactorily gone through postal tuition throughout the year are invited again for a second round of training (senior batch).
Stage 4: International Mathematical Olympiad (IMO)The team selected at the end of the camp, a "leader" and a "deputy leader," represent India at the IMO that is normally held in July in a different member country of IMO each year. The leader and deputy leader are chosen by NBHM from among mathematics teachers/researchers involved in the Mathematics Olympiad activity. IMO consists of two written tests held on two days with a gap of at least one day. Each test is of four-and-a-half-hours duration. Travel to IMO venue and return takes about two weeks. India has been participating in IMO since 1989. Students of the Indian team who receive gold, silver and bronze medals at IMO receive a cash prize of Rs. 5,000/-, Rs. 4,000/- and Rs. 3,000/- respectively, from NBHM during the following year at a formal ceremony at the end of the training camp.The Ministry of Human Resource Development (MHRD) finances international travel of the eight-member Indian delegation connected with international participation. NBHM finances the entire in-country programme and takes care of other expenditureStudents aiming for selection for participation in IMO should note that RMO is the first essential step for the programme. To appear for RMO, students should get in touch with the RMO co-ordinator of their region well in advance, for enrolment and payment of a nominal fee.Syllabus for Mathematics OlympiadsThe syllabus for Mathematics Olympiads (regional, national and international) is pre-degree college mathematics. The areas covered are: number systems, arithmetic of integers, geometry, quadratic equations and expressions, trigonometry, co-ordinate geometry, systems of linear equations, permutations and combinations, factorisation of polynomials, inequalities, elementary combinatorics, probability theory, number theory, infinite series, complex numbers and elementary graph theory. The syllabus does not include calculus and statistics. The typical areas for problems are: number theory, geometry, algebra and combinatorics. The syllabus is in a sense spread over class IX to class XII levels, but the problems under each topic are of an exceptionally high level in difficulty and sophistication.
Sunday, August 1, 2010
Matrix Algebra: Introduction
Matrices and Determinants were discovered and developed in the eighteenth and nineteenth centuries. Initially, their development dealt with transformation of geometric objects and solution of systems of linear equations. Historically, the early emphasis was on the determinant, not the matrix. In modern treatments of linear algebra, matrices are considered first. We will not speculate much on this issue.
Matrices provide a theoretically and practically useful way of approaching many types of problems including:
Solution of Systems of Linear Equations,
Equilibrium of Rigid Bodies (in physics),
Graph Theory,
Theory of Games,
Leontief Economics Model,
Forest Management,
Computer Graphics, and Computed Tomography,
Genetics,
Cryptography,
Electrical Networks,
Fractals.
Back to the Matrix Algebra page
Matrices provide a theoretically and practically useful way of approaching many types of problems including:
Solution of Systems of Linear Equations,
Equilibrium of Rigid Bodies (in physics),
Graph Theory,
Theory of Games,
Leontief Economics Model,
Forest Management,
Computer Graphics, and Computed Tomography,
Genetics,
Cryptography,
Electrical Networks,
Fractals.
Back to the Matrix Algebra page
Matrix
Matrix
A matrix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).trix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).
A matrix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).trix is an ordered set of numbers listed rectangular form.
Example. Let A denote the matrix
[2 5 7 8]
[5 6 8 9]
[3 9 0 1]
This matrix A has three rows and four columns. We say it is a 3 x 4 matrix.
We denote the element on the second row and fourth column with a2,4.
Square matrix
If a matrix A has n rows and n columns then we say it's a square matrix.
In a square matrix the elements ai,i , with i = 1,2,3,... , are called diagonal elements.
Remark. There is no difference between a 1 x 1 matrix and an ordenary number.
Diagonal matrix
A diagonal matrix is a square matrix with all de non-diagonal elements 0.
The diagonal matrix is completely denoted by the diagonal elements.
Example.
[7 0 0]
[0 5 0]
[0 0 6]
The matrix is denoted by diag(7 , 5 , 6)
Row matrix
A matrix with one row is called a row matrix
Column matrix
A matrix with one column is called a column matrix
Matrices of the same kind
Matrix A and B are of the same kind if and only if
A has as many rows as B and A has as many columns as B
The tranpose of a matrix
The n x m matrix A' is the transpose of the m x n matrix A if and only if
The ith row of A = the ith column of A' for (i = 1,2,3,..n)
So ai,j = aj,i'
The transpose of A is denoted T(A) or AT
0-matrix
When all the elements of a matrix A are 0, we call A a 0-matrix.
We write shortly 0 for a 0-matrix.
An identity matrix I
An identity matrix I is a diagonal matrix with all diagonal element = 1.
A scalar matrix S
A scalar matrix S is a diagonal matrix with all diagonal elements alike.
a1,1 = ai,i for (i = 1,2,3,..n)
The opposite matrix of a matrix
If we change the sign of all the elements of a matrix A, we have the opposite matrix -A.
If A' is the opposite of A then ai,j' = -ai,j, for all i and j.
A symmetric matrix
A square matrix is called symmetric if it is equal to its transpose.
Then ai,j = aj,i , for all i and j.
A skew-symmetric matrix
A square matrix is called skew-symmetric if it is equal to the opposite of its transpose.
Then ai,j = -aj,i , for all i and j.
The sum of matrices of the same kind
Sum of matrices
To add two matrices of the same kind, we simply add the corresponding elements.
Sum properties
Consider the set S of all n x m matrices (n and m fixed) and A and B are in S.
From the properties of real numbers it's immediate that
A + B is in S
the addition of matrices is associative in S
A + 0 = A = 0 + A
with each A corresponds an opposite matrix -A
A + B = B + A
Scalar multiplication
Definition
To multiply a matrix with a real number, we multiply each element with this number.
Properties
Consider the set S of all n x m matrices (n and m fixed). A and B are in S; r and s are real numbers.
It is not difficult to see that:
r(A+B) = rA+rB
(r+s)A = rA+sA
(rs)A = r(sA)
(A + B)T = AT + BT
(rA)T = r. AT
Sums in math
Because in the following, there is an intensive use of the properties of sums, the reader who is not familiar with these properties must read first Sums in math .
Remark. In this html document, for convenience, we'll write the word sum instead of the sigma sign.
Multiplication of a row matrix by a column matrix
This multiplication is only possible if the row matrix and the column matrix have the same number of elements. The result is a ordinary number ( 1 x 1 matrix).
To multiply the row by the column, one multiplies corresponding elements, then adds the results.
Example.
[1]
[2 1 3]. [2] = [19]
[5]
Multiplication of two matrices A.B
This product is defined only if A is a (l x m) matrix and B is a (m x n) matrix.
So the number of columns of A has to be equal to the number of rows of B.
The product C = A.B then is a (l x n) matrix.
The element of the ith row and the jth column of the product is found by multiplying the ith row of A by the jth column of B.
ci,j = sumk (ai,k.bk,j)
Example.
[1 2][1 3] = [5 7]
[2 1][2 2] [4 8]
[1 3][1 2] = [7 5]
[2 2][2 1] [6 6]
[1 1][2 2] = [0 0]
[1 1][-2 -2] [0 0]
From these examples we see that the product is not commutative and that there are zero divisors.
Properties of multiplication of matrices
Associativity
If the multiplication is defined then A(B.C) = (A.B)C holds for all matrices A,B and C.
Proof:
We'll show that an element of A(B.C) is equal to the corresponding element of (A.B)C
First we calculate the element of the ith row and jth column of A(B.C)
Let D denote B.C, then
dk,j = sump bk,p.cp,j (1)
Let E denote A.D then
ei,j = sumk ai,k.dk,j (2)
(1) in (2) gives
ei,j = sumk ai,k.(sump bk,p.cp,j)
<=> ei,j = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of A(B.C) is
sumk,p ai,k.bk,p.cp,j (3)
Now we calculate the element of the ith row and jth column of (A.B)C
Let D' denote A.B, then
di,p' = sumk ai,k.bk,p (4)
Let E' denote D'C then
ei,j' = sump di,p'.cp,j (5)
(4) in (5) gives
ei,j' = sump (sumk ai,k.bk,p).cp,j
<=> ei,j' = sumk,p ai,k.bk,p.cp,j
So the element of the ith row and jth column of (A.B)C is
sumk,p ai,k.bk,p.cp,j (6)
From (3) and (6) => A(B.C) = (A.B)C
Distributivity
If the multiplication is defined then A(B+C) = A.B+A.C and (A+B).C = A.C+B.C holds for all matrices A,B and C. This theorem can be proved in the same way as above.
Theorem 1
For each A, there is always an identity matrix E and an identity matrix E' so that A.E = A and E'.A = A If A is a square matrix, E = E'.
Theorem 2
(A.B)T = BT .AT
This theorem can be proved in the same way as above.
Theorem 3
If the multiplication is defined then for each A
A.0 = 0 = 0.A
Theorem 4
r and s are real numbers and A , B matrices. If the multiplication is defined then (rA)(sB) = (rs)(AB) This theorem can be proved in the same way as above.
Theorem 5
if D = diag(a,b,c) then D.D = ( a2 , b2 , c2)
D.D.D = ( a3 , b3 , c3)
.....
This property can be generalised for D = diag(a,b,c,d,e,...,l).
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