ttmo syllabus
DESCRIPTION
Training syllabus of the Trinidad and Tobago Mathematics olympiad.TRANSCRIPT
Contents
1 Introduction 3
2 A Brief History of the TTMO 3
3 The IMO 33.1 What is the IMO? . . . . . . . . . . . . . . . . . . . . . . . 33.2 Examination structure . . . . . . . . . . . . . . . . . . . . . 43.3 Award of medals . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Regulations . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 Training Schedule 54.1 First stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 Weekly training . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Training camp . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 List of Topics 65.1 Logical Preliminaries . . . . . . . . . . . . . . . . . . . . . 75.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 85.4 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . 95.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . 10
6 Lecture Plan 116.1 January–June . . . . . . . . . . . . . . . . . . . . . . . . . 116.2 September–December . . . . . . . . . . . . . . . . . . . . . 116.3 Camp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.4 A note about returning students . . . . . . . . . . . . . . . 11
7 References and Training Material 12
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1 Introduction
A more colourful introduction will appear later.This document attempts to fill the structural lacuna that currently exists
in the training regime of The Trinidad and Tobago Mathematics Olympiad(TTMO).
To ensure that this document is viewed its proper historical context orsomething of the sort, its first section contains a review of the evolution ofthe TTMO.
The following section describes briefly the format of the InternationalMathematics Olympiad (IMO), participation in which is the ultimate goalof any aspiring young mathematician.
Section 4 outlines the path that a student from Trinidad and Tobago musttake in order to compete at the IMO.
Sections 5 and 6 form the core of the document.A list of references and training material is provided in Section 7.
2 A Brief History of the TTMO
Coming soon.
3 The IMO
3.1 What is the IMO?
The raison d’être of the TTMO is the International Mathematics Olympiad(IMO).
Taken from www.imo-official.org:
“The IMO is the World Championship Mathematics Competi-tion for High School students and is held annually in a differentcountry. The first IMO was held in 1959 in Romania, with 7countries participating. It has gradually expanded to over 100countries from 5 continents. The IMO Advisory Board ensuresthat the competition takes place each year and that each hostcountry observes the regulations and traditions of the IMO.”
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The contest is usually held in the second week of July each year, andis open to students below nineteen years of age. A participating countrymay send a team of at most six students. In addition to the competitors,the delegation must include a Team Leader and a Deputy Team Leader. Acountry may also elect to have one or more Observers accompany the team.These members usually comprise a subset of those responsible for the trainingand preparation of the team; their roles are outlined in the General Regulationsof the IMO (see Subsection 3.4).
3.2 Examination structure
The examination consists of six problems worth seven points each, the totalscore thus being forty-two points. The examination is held over two consecu-tive days; the contestants have four-and-a-half hours to solve three problemsper day, with each contestant working individually.
The problems are chosen from various areas of High School mathemat-ics, broadly classifiable as Algebra, Geometry, Number Theory, and Combi-natorics. The solutions of these problems require no knowledge of highermathematics such as Calculus or Abstract Algebra, and are often short andelementary, yet elusive.
3.3 Award of medals
The total number of medals (gold, silver, and bronze) does not normallyexceed half the total number of contestants. The numbers of gold, silver, andbronze medals awarded satisfy (approximately) the ratio 1:2:3 respectively.Special prizes may be awarded for outstanding solutions. Each contestantwho has not received a gold, silver, or bronze medal receives a Certificate ofHonourable Mention if he or she has received seven points for the solution ofat least one problem.
3.4 Regulations
For details regarding the material discussed in Subsections 3.1–3.3 and muchmore, the reader is referred to the General Regulations of the IMO which canbe found at:
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http://www.imo-official.org/documents/RegulationsIMO.pdf.
Prospective delegation members are urged to familarise themselves withthe rules and provisions stipulated in the General Regulations.
4 Training Schedule
Contained in this section is an overview of the stages of the yearly competitionheld by the TTMO.
4.1 First stage
The first stage of the competition is split over two rounds of examinations.The Round One contest is held in September and is open to all studentsfrom Forms Two to Six. There are three categories of competition: Junior,Intermediate, and Senior.
The Round Two contest is held the following January, and there are twocategories of competition: Level I and Level II. Students who have excelledin the Round One competition are allowed to sit the examinations of RoundTwo, with those from Forms Two to Five competing at Level I, and thosefrom Form Six competing at Level II. Students who excel in the Round Twocompetition are invited to attend weekly training sessions with the view tobeing selected to represent Trinidad and Tobago at the IMO.
4.2 Weekly training
The students who are chosen based on their performance in the Round Twocompetition attend one three-hour training session per week during the Hilaryand Trinity terms (January–June), where they are lectured in areas of prob-lem solving and advanced High School mathematics. Several selection testsare administered, and using the results of these, a suitable team is chosen.It should be noted here that, while a maximum of six students per team ispermitted by the rules of the IMO, a participating country may opt to sendless than six students, should they deem only those students fit for the com-petition. The TTMO is guided in this manner and may choose to form ateam of less than six students; through the selection examinations each team
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member must prove that he or she has attained the standard necessary tocompete meaningfully at the IMO.
Returning students who are still eligible to compete at the following year’sIMO are invited to continue attending weekly training sessions in Michaelmasterm (September–December).
4.3 Training camp
The team is required to attend a two-week training camp which takes placeimmediately before they are set to travel to host country of the IMO. Campis an invaluable component of the training regime, and provides a period ofintense preparation before competition.
5 List of Topics
This section, together with the next, form the core of the document. Itcontains a list of the topics that the TTMO attempts to cover in a yearlongcycle.
A student attempting the IMO should have a working knowledge of thematerial outlined in this section. Two things should be noted. Firstly, thislist is not exhaustive in the context of olympiad mathematics, nor does itattempt to be. It is simply a syllabus that has been constructed by inspect-ing past examinations and consulting various olympiad training manuals (seeSection 7). As a corollary of this, one should expect that this section will berevised frequently to reflect changes in examination trend.
There is a caveat. Listed below are certain topics which, although theymerit a permanent position in any similar syllabus, have fallen out of favour inrecent olympiads. The topic of Algebraic Inequalities is a current example ofthis anomaly. Problems involving an inequality which could be proven usingpurely algebraic methods were very common in olympiads pre-2009, but theirpopularity has waned over recent years. (Lovers of inequalities should be notafraid, for such an acute dip in density is usually short-lived.) To suggest thatthe topic of Algebraic Inequalities be removed (or even temporarily suspended)from this list is to ignore their fundamental position in the theory. Rather,the author hopes that those charged with training the students of the TTMO
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keep abreast with sharp shifts examination trends, and tailor their lecturesaccordingly.
Secondly, a “working knowledge” of the topics listed in this section is by nomeans a guarantee of exceptional performance at the IMO. (Statistical evi-dence supports the truth of the converse however.) This syllabus is worthlessif not accompanied by the appropriate training material, which must includea wide range of problems with varying levels of difficulty.
5.1 Logical Preliminaries
• Statements
Mathematical statements. Construction of statements. Conditionalstatements and implication. The converse and contrapositive of a con-ditional statement. The meaning of “assume without loss of generality”.
• Proof techniques
The concept of mathematical proof. Direct proof. Proof by contradic-tion. Proof by induction. Examples of fallacious proofs.
5.2 Algebra
• Functions
Domain and range. Injectivity and surjectivity. The inverse of a func-tion. Even and odd. Periodicity. Some special functions: modulus,floor, ceiling. Transforming functions. Solving functional identities.
• Equations and Expressions
Manipulation of algebraic expressions. Special factorisations: sums and
differences of nth powers. Expressions of the form xn`1
xn. Polynomials:
multiplying and dividing polynomials, roots of polynomials, coefficientsand roots, transforming polynomials, Newton’s Sums. Symmetric ex-pressions. Systems of equations: linear systems and Gaussian elimina-tion, convenient systems.
• Sequences and Series
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Explicitly defined sequences. Recursively defined sequences. Arithmeticseries. Geometric series. Infinite series. Telescoping series. Solv-ing recurrences. Computing sums and products. Harmonic sequences.Continued fractions.
• Inequalities
Absolute value inequalities. Squares are positive. The Arithmetic Mean–Geometric Mean Inequality. The Quadratic Mean. The HarmonicMean. Weighted means. The Cauchy–Schwarz Inequality. Maximisa-tion and minimisation. The Rearrangement Inequality. Chebyshev’s In-equality. Convexity and Jensen’s Inequality. Hölder’s Inequality. Bernoulli’sInequality. Minkowski’s Inequality. Schur’s Inequality. Muirhead’s In-equality. Homogenisation. The Ravi Transformation. Geometric In-equalities.
5.3 Geometry
• Angles
Angles and parallel lines. Arcs, segments, sectors and angles. Anglesformed by lines intersecting a circle.
• Triangles
Medians, angle bisectors, perpendicular bisectors, altitudes. Congru-ent triangles. Similar triangles. The Angle Bisector Theorem. Righttriangles and the Pythagorean Theorem. The trigonometric functions.The Sine Rule. The Cosine Rule. The area of a triangle. Stewart’sTheorem.
• Quadrilaterals
Trapezoids. Parallelograms. Rhombi. Rectangles and squares. Cyclicquadrilaterals: properties of cyclic quadrilaterals, finding cyclic quadri-laterals, Ptolemy’s Theorem.
• Polygons
Types of polygons. Angles in a polygon. Regular polygons.
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• Area
Similar figures. Same base or same altitude. Complicated figures.
• Circles
The Power of a Point. The Radical Axis.
• Collinearity and Concurrency
Menelaus’ Theorem. Ceva’s Theorem. Desargues’ Theorem. Jacobi’sTheorem.
• Transformations
Translation. Rotation. Reflection. Distortion. Dilation and homothecy.Projection. Inversion.
• Analytic Geometry
Labelling the plane. Lines, angles, and distances. Parameters. Vectors.Complex numbers and the Argand diagram.
5.4 Number Theory
• Exponents and Logarithms
Integer and rational exponents. Simplifying radical expressions. Ratio-nalising denominators. Logarithms.
• Proportions
Direct and inverse. Manipulating proportions. Conversion factors.
• Number bases
Working with different bases. Uniqueness of representation.
• Divisibility
Division with remainder. Divisibility properties. Divisibility tricks. Primenumbers and composite numbers. The infinitude of the primes. Decom-position as a product of prime factors. The Fundamental Theorem ofArithmetic. The greatest common divisor and least common multiple.The Euclidean Algorithm. Relatively prime and pairwise relatively primesets of numbers.
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• Congruences
Modular arithmetic. Residue classes. Linear congruences. Linear Dio-phantine equations. The Chinese Remainder Theorem. Quadratic con-gruences and quadratic residues. The Legendre Symbol. Quadraticreciprocity. Fermat’s Little Theorem. Wilson’s Theorem.
• Arithmetical functions
Multiplicative functions. Euler’s φ-function. The Euler–Fermat Theo-rem. The number of divisors. The sum of the divisors. Perfect numbers.Deficient numbers and abundant numbers.
• Diophantine equations
Pythagorean triples. The equation x4 ` y 4 “ z2. The Pell Equation.The Four-Square Theorem.
• Prime numbers
The role of the primes. Primes of the form 4k`1, and those of the form4k `3. Dirichlet’s Theorem. The difference between primes. Reclusiveprimes and twin primes. The sum of the reciprocals of the primes. Thedistribution of the primes. The Goldbach Conjecture1.
5.5 Combinatorics
• Counting principles
The Pigeonhole Principle. Multiplication. Restrictions on multiplica-tion. The Inclusion-Exclusion Principle. Generating functions. Parti-tions. The Invariance Principle. Infinite sets and one-to-one correspon-dences.
• Permutations and combinations
Permutations. Combinations. Rearrangements and derangements. TheBinomial Theorem. Pascal’s Triangle. Combinatorial identities.
• Graphs1At the time of writing this is still a conjecture.
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Edges, vertices, and degrees. Planar graphs. Paths, trails, and cycles.Colourings.
6 Lecture Plan
It is the intention of the author to produce a detailed lecture plan for eachtraining period (January–June, September–December, and camp) to whichtrainers should adhere. However the syllabus is new, and he will most certainlycomplete this task far more efficaciously when equipped with the hindsightgained by attempting to lecture the syllabus detailed in Section 5 over thenext year. In short, watch this space.
6.1 January–June
Nothing yet. Stay tuned.
6.2 September–December
Nothing yet. Stay tuned.
6.3 Camp
At this point the students should be attempting difficult problems from eachof the four subjects. Advanced and alternative topics may be taught accordingto taste.
6.4 A note about returning students
An obvious question arises: How does this lecture plan and syllabus makeallowances for returning students who have already completed a full year oftraining with the TTMO?
The author offers the following reasons as to why returning studentsshould continue attending weekly training sessions.
(i) To expand their existing knowledge with new material arising from majorchanges in the syllabus.
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(ii) To fill any gaps in their existing knowledge.
(iii) To revise difficult concepts that they may not have fully mastered ontheir first pass.
(iv) To consolidate their existing knowledge with minor changes in the syl-labus.
(v) To seek the expertise of trainers on advanced or optional topics thatthey may be reading for self-study.
Should the trainer deem appropriate, during a weekly session, he or shemay group senior students and conduct an advanced class in parallel with theusual lecture.
At the committee’s discretion, seasoned IMO competitors may be allowedto be absent from weekly training sessions. However, all students wishing torepresent Trinidad and Tobago at the IMO must sit the immediately precedingTTMO selection examinations.
7 References and Training Material
There exists a plethora of resources available both in print and on the worldwide web, and the niche area of Olympiad Mathematics is an active one. Oneshould therefore expect this section of the document to be ever-expanding,and the author welcomes additions from fellow trainers.
• http://www.imo-official.org
• http://www.artofproblemsolving.com
• Paul Erdös and Janos Surányi. Topics in the Theory of Numbers.Springer-Verlag, 2003
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