# From the Mathematics Faculty

Dear Trinity Mathematicians,

You are invited to participate in the 2021 Mathematics Challenge for Young Australian (MCYA) programme. This is a problem-solving programme organised by the Australian Mathematics Trust (AMT), which is generally aimed at the top 15% of students in an academic year at the School.

Participants will be required to present written solutions to as many of problems set in either Stage. The problems require time and persistence and may be based on Mathematics studied to date or learned as you progress through the Stage/Series problems.

**The Challenge Stage:**

The MCYA Challenge Stage will take place over a continuous four-week period commencing later this term (and continuing in the forthcoming vacation period). Participants will receive a problem book containing six questions (in either the Junior Division for Years 7 and 8 and the Intermediate Division for Years 9 and 10).

Participants are advised of the following dates:

- Date of issue of materials will be on or before Monday March 22
- Due date for submission of solutions to attempted problems will be on or before Monday April 26.

Confirmed entrants will have access to a dedicated Mathematics Competitions Canvas Course and further instructions will be notified soon after registration. In this Series, participants **may work with a partner** on any of the problems in the same Division although **individual solutions must be submitted. **Details regarding submission of solutions and further information will be made available on the Mathematics Competitions Canvas Course.

**The Enrichment Stage:**

The MCYA Enrichment Stage will take place over a continuous 12-week period commencing in June through to August, 2021. Depending on the Series selected (Dirichlet, Euler, Gauss, Noether, Polya), participants will receive a problem book containing 8 to 16 questions based on topics in Number, Algebra, Geometry, Measurement and Problem Solving.

Participants are advised of the following dates:

- Date of issue of materials will be on or before Tuesday June 1
- Due date for submission of solutions to attempted problems will be on or before Tuesday August 17.

Like the Challenge Stage, confirmed entrants will have access to a dedicated Mathematics Competitions Canvas Course and further instructions will be notified soon after registration.

Each question in an Enrichment problem booklet is based on Mathematics contained in a comprehensive booklet of theory notes, examples and exercises. These notes, examples and exercises should be consulted **before** completing any of the related questions in the actual problem booklet. Sample questions and corresponding solutions will be provided on the Mathematics Competition Canvas Course.

In this Series, collaboration among other entrants is **NOT** permitted.

A brief description of each Enrichment Series is presented below.

**The DirichlĂ©t Enrichment Series**

This Series is suitable for students in Year 7. Three of the eight chapters revolve around a story which illustrates some problem-solving techniques: using logic, solving a simpler problem, and working backwards. The other five chapters cover:

- Tessellations
- One-handed arithmetic
- Time, distance, speed
- Working with patterns
- Recurring decimals

Almost all the material presented is Mathematics not usually taught in school at any level and so is appropriate for enrichment purposes. The Student Problems book has eight questions.

**The Euler Enrichment Series**

This Series is mainly for Year 8 and outstanding Year 7 students. The topics considered in this series are:

- Primes and composites
- Least common multiples
- Highest common factors
- Arithmetic sequences
- Figurate numbers
- Congruences
- Properties of angles
- Counting techniques
- The Pigeonhole Principle

Chapters 1 to 4 involve very little algebra, and the latter chapters will require students to learn about or have more advanced Algebra skills. The Student Problems book has 12 questions.

**The Gauss Enrichment Series**

This is designed for talented students in Years 8 and 9. It introduces the use of computer spreadsheets such as Excel. The topics covered include:

- Parallels
- Similarity
- Pythagorasâ€™ Theorem
- Spreadsheets
- Diophantine equations
- Counting techniques
- Congruence

Each chapter introduces new Mathematics and presents some key mathematical ideas followed by some illustrative examples with suggested approaches and sets of exercises for which there are fully worked solutions. The Student Problems book has twelve questions.

**The Noether Enrichment Series**

This is designed for talented students in Years 9 and 10, following on from Gauss Series. The topics considered are:

- Expansion and factorisation
- Sequences and series
- Number bases
- Inequalities
- Methods of proof
- Congruence
- Circles
- Tangents

Spreadsheets may also be useful for some problems. The Student Problems book has sixteen questions.

**The Polya Enrichment Series (by invitation ONLY)**

This Series extends the work from earlier Series on Euclidean geometry. There is an introduction to some selected topics in advanced algebra. The topics covered are:

- Functions
- Symmetric Polynomials
- Geometry
- Inequalities
- Functional Equations
- Number Theory
- Counting
- Graph Theory

It would be useful for students to compile an ongoing summary of the facts and techniques learned and use them to obtain their own solutions to the examples. The Student Problems book has 16 problems.

For both the Challenge and Enrichment Stages, the AMT has indicated that students can seek help from any resources **other** than human ones, so textbooks, encyclopaedias, etc., calculators and computers are suitable. ** Websites offering help with problems are really human resources and are not allowed**, but the use of a computer-based encyclopaedia or a website providing information of a general nature is alright.

Supervised support for participants in this programme will be available across two lunch sessions in a week. The time and venue of these support sessions will be confirmed with the participants on the Mathematics Competitions Canvas Course. Some guidance on how the solutions are to be prepared and submitted will also be shared with confirmed participants.

If accepted, Trinity Grammar School will absorb the cost of entry of participants into either Stage.

We hope the MCYA Challenge and/or the Enrichment Stages is an enjoyable and valuable learning experience for participants.

*If you wish to enter any of the Stages (Challenge and/or Enrichment), please click on the following link(s) to register your interest no later than Monday 8 March 2021. Unfortunately, we are unable to register submissions after this date*

If you have any questions, please contact Miss Cho (Senior Mathematics Teacher) at hcho@trinity.nsw.edu.au.

**Hyun Mi Cho **| Teacher | Secondary [Math] | MiC Math Club

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