Mathematics

Solve this problem

My cat gets on the roof of our house by jumping first to the fence, then on to the water tank, then on to the roof of the shed, then on to the pergola and finally on the roof. However, coming down, she can omit as many of the intermediate steps as she wishes. How many routes can my cat take coming down? (2000 Australian Mathematics Competition, Junior question 29) » See the solution

 

‘Mathematics is the most beautiful and most powerful creation of the human spirit’ – Stefan Banach 1892–1945, Polish mathematician

 

The Australian Mathematics Trust (AMT) is a national not-for-profit organisation whose purpose is to enrich the teaching and learning of mathematics for students of all standards. Our vision is that all young Australians should have the opportunity to realise their intellectual potential in mathematics. We conduct a number of mathematics competitions and enrichment programs throughout schools in Australia and in nearly 40 countries around the globe. The AMT is the only organisation providing a complete pyramid of enrichment programs, ranging from suitable for the average student to providing the only pathway to representing Australia at International Olympiads in mathematics and informatics. We hold open events for all students and invitational competitions for identifying exceptional mathematical talent. Our competitions and enrichment programs focus on problem solving as essential in unlocking a student’s success in the understanding of mathematics. All of our activities feature original problems created by Australia’s leading mathematics teachers and academics.

 

Australian Mathematics Competition

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The Australian Mathematics Competition (AMC) is one of the largest single events on the Australian education calendar. Now in its 38th year, the competition was a first in Australia and remains one of the biggest mathematics competitions of its kind in the world. It has attracted more than 14.5 million entries since its inception in 1978, with multiple generations now having participated. Primary and secondary school students take part annually, attempting the same problems on the same day. It has become a truly international event, with participation by countries throughout Europe, the Middle East, Africa, the Pacific and South East Asia. The AMC is for students of all standards. Students are asked to solve thirty problems in 60 minutes (Years 3–6) or 75 minutes (Years 7–12). The earliest problems are very easy and all students should be able to attempt them. The problems get progressively more difficult until the end, when they are challenging to the most gifted student. All students receive a certificate and a detailed report showing how they went on each problem, with comparative statistics. Prizes and medals are awarded at annual state presentations in early November. The competition is designed to attract the average student to the study of mathematics by demonstrating its place in everyday life and making it fun. It also identifies and opens the way for further nurturing of the exceptionally talented. Most of Australia’s leading mathematicians aged below 40 were identified and developed as a result of taking part in the competition.

» More information on the AMC  |  » How to enter the AMC

 
 
 
 

Mathematics Challenge for Young Australians

March–September

The Mathematics Olympiads are supported by the Australian Government Department of Education through the Mathematics and Science Participation Program.

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The Mathematics Challenge for Young Australians (MCYA) is a staged program designed to help teachers motivate, stimulate, encourage and develop mathematically interested students in Years 3 to 10 to bring forth the talent and potential within. Teachers find the materials valuable, allowing them to better assist the students in their care. The MCYA program may be particularly useful in schools where teachers work in isolation and have a handful of talented students spread out over a number of classes.

Challenge Stage (March–June)

The CHALLENGE Stage is an opportunity for students who enjoy mathematics to develop their problem-solving skills. It consists of four problems for Years 3 and 4 students (Middle Primary) and Years 5, 6 and 7 students (Upper Primary), six problems for Years 7 and 8 students (Junior) and six problems for Years 9 and 10 students (Intermediate). Teachers can select a consecutive three-week period between March and June for students at their school to participate. Detailed support notes, solutions and extension problems are provided to teachers.

Enrichment Stage (April–September)

The ENRICHMENT Stage consists of six different parallel programs run over a six-month period with comprehensive student and teacher support materials. Each of the six programs is aimed at different levels of students in Years 5 to 10. It is an opportunity for talented students who enjoy mathematics to formally extend their knowledge of mathematics with course work that augments the school curriculum.

AIMO – Australian Intermediate Mathematics Olympiad (6 September)

Schools that participate in the MCYA Challenge and Enrichment Stages are invited to enrol interested students, seeking a further mathematics challenge, in this 4-hour exam in September. High achievers in the AMC are also invited to enter and be challenged with a range of interesting questions. The AIMO is one of the competitions used to identify students who may be offered special enrichment classes and other opportunities to enhance their enjoyment and knowledge of mathematics. » Enter now

» More information on MCYA | » Challenge Practice | » Enrichment Practice

 
 
 
 

Australian Mathematical Olympiad Committee Invitational Program

All Year

The Australian Mathematical Olympiad Committee (AMOC) Invitational Program is a program which offers  the top students from the Mathematics Challenge for Young Australians and the Australian Mathematics Competition  extra enrichment opportunities, and can lead to selection in the Australian team to participate in the International Mathematical Olympiad (IMO). The IMO is the pinnacle of competition between students of pre-university level from different countries. For more information about our informational program click on the link below.

» More information on the AMOCIP | » IMO Results

 

Australian Statistics Competition

Registration closes early August, entries due beginning of September

Supported by the Australian Bureau of Statistics

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The Australian Statistics Competition (ASC) is an activity that assists teachers to present lessons on data and statistics in a relevant and relaxed environment. The chance and data stream of school mathematics syllabi is particularly well suited for practical, project-based and team-based learning activities, and the competition offers a framework and focus to support such learning activities.

» More information about the ASC | » 2015 ASC Results

 

Solution

There are four intermediate steps from the roof to the ground, so, from any position above the ground, the cat has the choice to use or not use the next intermediate step, i.e. two choices in each case (except when on the fence, where there is only one way). This gives 2 x 2 x 2 x 2 = 24 = 16 different ways to reach the ground, as shown by the tree diagram below, where each branch splits into two for each of the two choices. Mathematics Answer Diagram If you enjoyed this problem, visit our Shop for more past problems and solutions.