1.1 Shortcuts in Computation
1. Quicker Counting Methods 
Grouping numbers that add up to 5 or 10 
73+74+27+26 =(73+27)+(74+26) 

Round off numbers that are close to 5 or 10 
73+74+27+26 =752+751+25+2+25+1 
2. Sum of numbers that form a pattern 
For patterns where: numbers increase/decrease by same value 
1. rewrite sum in reverse order underneath 2. pair up and sum 3. sums of pairs are the same 4. Since sums are the same, multiple sum by number of pairs 5. Divide by 2 

Example: 
Find 2+4+6+...+78+80
2+4+6+...+78+80 80+78+...+6+4+2 82*40=3280 3280/2=1640
2+4+6+...+78+80=1640 
3. Quicker Muliplication Methods 
Remember numbers in their expanded form 
3526=3000+500+20+6 

3.1 Multiples of 10 
30x25 =3x10x25 =3x250 =750 

3.2 Multiples of 5 
25x6 =5x5x6 =5x30 =5x3x10 =15x10 =150 
1.2 Number Logic
Properties of Numbers 
Primes 
factor of 1 and itself only 

Composites 
factors other than itself 
Divisbility Rules 
Divisibility rule of 2 
EVEN ends with 0,2,4,6,8 

Divisibility rule of 3 
sum of its digits divisble by 3 

Divisibility rule of 4 
last 2 digits divisible by 4 

Divisibility rule of 5 
ends with 0 or 5 

Divisibility rule of 6 
EVEN AND divisible by 3 

Divisible by 8 
last 3 digits divisible by 8 

Divisible by 9 
sum of its digits divisble by 9 

Divisible by 10 
ends with 0 
Squared Numbers 
NxN=N^{2} 
eg 2x2=2^{2}=4 
Cubed Numbers 
NxNxN=N^{3} 
eg 2x2x2=2^{3}=8 
1.3 Developing Patterns and Shortcuts
Factorising Numbers 
A number is factorised when expressed as a product of prime numbers 
250 =2x125 =2x5x25 =2x5x5x5 

Find prime factors 
HCF 
largest counting number that divides into both exactly 
Highest Common Factor 
method 
1. factorise 2. multiply the factors that are common only those factors that have a pair 

example 
HCF of 240 and 924 240=2x5x2x2x3x2 924=3x2x7x11x2 HCF=2x3x2=12 
LCM 
Of all the multiples of the 2 numbers, its the smallest multiple they have in common 
Lowest Common Multiple 
method 
1. factorise 2. multiply the factors that are common and factors they dont have in common 

example 
LCM of 120 and 140 120=2x2x2x3x5 140=2x2x3x7 LCM=2x2x3x2x5x7 
Question (find multiples) 
Jack, Art, Fran and Megan work as volunteers at the local kennel. Jack gives the dogs baths every 4 days. Art cleans out cages every 6 days. Frand feeds the animals in section b every 2 days. Megan helps the receptionist every 3 days. How many times in 12 weeks will all 4 helpers be at the clinic on the same day? 
how to solve Find all the common multiples from 6 days to 84 days (12 weeks) of 4, 6, 2, 3 
Question (LCM) 
Two buses leave the terminal at 8am. Bus A takes 60mins to complete its route and Bus B takes 75mins. When is the next time the two buses will arrive together at the terminal (if they are on time)? 
how to solve 1. Find LCM of 60 and 75. 2. Add LCM to 8am 
Question (HCM and LCM) 
Dennis has a choice between two house numbers on Small Street. The two house numbers have their highest common factor of 6. Their least common multiple is 36. One of the house numbers is 12. What is the other number? 
how to solve work backwards 
1.4 Logic Deduction
Logic Deduction Problems 
If need to add groups of things, use biggest numbers first 
Question 
What is the minimum number of coins needed to make $4.85 from only 5c, 20c, 50c coins 
Start with biggest coins first, working through to smaller coins 9x50c=$4.50 1x20c=20c 3x5c=15c 

assume worse case scenario 

investigate standard case 

write relations between numbers down 
1.5 Space, Area and Volume
Area of Rectangle 
length x width 
Area of Triangle 
A = base x height / 2 
Volume of Cube 
V = a^{3} where a is length of a side 
Volume of Rectangular Prism 
V = length x height x depth 
1m 
= 100cm 
Finding Area of Rectangular Shapes 
Method 1 
Divide shape into rectangles 
Find area of each and find total 
Method 2 
Extend shape into one larger rectangle 
1. Find area of larger rectangle (X) 2. Find area of missing rectangle (Y) 3. Larger rectangle (X)  Missing rectangle (Y) 
1.6 Equations
Pronumerals 
Boxes to store missing numbers 

Letters to represent unknown numbers 

Use x, y and z 
Rearranging Equations 
= is like a balancing scale 

solving an equation 
aim of finding the unknown number 

rearranging equations 
how to solve an equation 


how if we do something to one side, we need to do the same thing to the other side 


eg. if we add 3 to one side, we need to add 3 to the other side 


eg. if we times by 3 to one side, we need to times by 3 to the other side 

+ 
 

x 
/ 
Simultaneous Equations 
if there are 2 unknowns, need 2 equations 
1. Solving by Adding and Subtracting Equations 
example 
5x  y = 4 (1) 2x + y = 10 (1)+(2) 7x = 14 x = 2 y = 6 

example 
7x + y = 18 (1) 2x + 2y = 12 (2) (1) x 2 14x + 2y = 36 (1a) (1a)  (2) 12x = 24 x = 2 y = 4 
2. Solving by Substitution 
method 
1. rearrange one equation for y 2. substitute y into other equation 

example 
5x  y = 4 (1) 2x + y = 10 (2) rearrange (1) y = 5x  4 (1a) substitute (1a) into (2) 2x + (5x  4) = 10 x = 2 y = 6 
Turning word problems into an equation 
Step 1 
What are the unknowns? 
Give each a letter, x, y 
Step 2 
Find the equations to solve 
Step 3 
Solve the simultaneous equations 
Example Questions 
The quotient of two numbers is 4 and their difference is 39. What is the smaller number of the two 
The sum of the ages of Alan and Bill is 25; the sum of the ages of Alan and Carl is 20; the sum of the ages of Bill and Carl is 31. Who is the oldest of the three boys and how old is he? 
1.7 Probability, Venn Diagrams and Whodunits
1. Certainty Problems 
Typical Question 
Suppose that there are ten black and ten navy socks in your drawer. Your room is dark and you cannot turn on the light. What is the smallest number of socks that you must take out of your drawer to be certain that you have a pair of the same colour? 
Basically, to be certain of "an outcome", what is the smallest number of "actions" required to take 
Strategy 
Start from smallest and go up 

1 sock 
can't be certain 

2 socks 
can't be certain 

3 socks 
can be certain 
2. Certainty Problems with Restrictions 
Typical Question 
As above question, but what is the smallest number of socks needed to ensure we get a pair of black socks 
Restriction is it must be black socks 
Strategy 
Think Worst Case Scenario 

Worst case is you could in 10 picks, pick only Navy socks. 2 more picks you'll be certain to get a pair of black socks 

12 socks 
can be certain 
Venn Diagrams 
circle represents sets or groups of things that are same 
Example Question 
There are 160 students in Year 5. Of these students, 69 walked to school and 57 caught a train to school. If 148 students either walked to school or caught the train, how many students walked and caught a train to school? 

Draw a Venn diagram with a circle for students that walked and students that caught the train 

Where they overlap, are the number of students that walked and caught the train 
Whodunits 
Strategy 
Use a table, with different characteristics in columns and members of a group in rows 
Usually the answer needed are the characteristics 
Example Question 
Martin, Bill and Dave (members of a group) play first base, second base, and third base (characteristics) on their school softball team, but not necessarily in that order. Martin and the third baseman took Dave to the movies yesterday. Martin does not play first base. Who's on first base? 


1.8 Motions, Books, Clocks and Work Problems
1. Motion Problems 
distance 
= rate x time 
Example Question 1 
Two trains leave the same station at the same time, but in opposite directions. One train averages 56 km/h and the other averages 64 km/h. How far apart will the trains be when three hours have passed? 
Strategy 
Step 1 
Whats the distance after 1hr? (Draw a diagram) 


56km + 64km = 120km 


56km/hr + 64km/hr = 120km/hr 

Step 2 
Whats the distance after 3hrs? 


120km x 3 = 360km 

if opposite direction, 
add 
Example Question 2 
Suppose that these two trains start from the same station at the same time, this time in the same direction. How far apart will the fronts of the trains be at the end of the three hours? 

Step 1 
Whats the distance after 1hr? (Draw a diagram) 


64km/hr  56km/hr = 8km/hr 


64km  56km = 8km 

Step 2 
Whats the distance after 3hrs? 


8km x 3 = 24km 

if opposite direction, 
subtract 
2. Book Problems 
look at the structure of counting numbers used for book pages 
Example Question 
A printer uses an oldstyle printing press and needs one piece of type for each digit in the page numbers of a book. How many 2s will the printer need to print page numbers from 1 to 250 
consider the numbers place by place 

number of times 2s appear in the 1s place 
25 

number of times 2s appear in the 10s place 
30 

number of times 2s appear in the 100s place 
51 

answer 
=25+30+51 =106 
3. Clock Problems 
elapse time 
amount of time that has passed 
solve using 
facts about time 
Example Question 
A certain clock gains one minute of time every hour. If the clock shows the correct time now, in how many hours will it next show the correct time again without regard to am or pm? 
Fact 1 
A clock that has stopped 
Will show the correct time every 12 hrs. As it stopped at 6.03am on Monday. It was correct at the time it stopped. It will be correct again, when the time is at 6.03pm 
Fact 2 
The clock in the problem must gain 12 hours to show correct time again 
thus 
12 hrs 
=60mins x 12 = 720mins

thus 
as clock gains 1 min in 1hr 
the clock will gain 720min in 720hrs 


720/24=30days 
4. Work Problems 
solving using 
fractional parts of whole numbers and draw diagrams 
Example Question 
Paul can do a certain job in 3hrs and John can do the same job in 2hrs. At these rates, how long would it take Paul and John to do this job if they work together 
Strategy 
Step 1 
Draw a diagram for Paul and John. Fractional parts done in each hour 

Step 2 
Using the diagram, in one hour they can complete 1/3 + 1/2 = 5/6 of the job 

Step 3 
Work out how long to complete job 


1/5 of job left 


60min / 5 = 12mins to complete 1/5 of job 

answer 
=1hr 12mins 
1.9 Problem Solving Strategies
1. Drawing a picture or diagram 
Example Question 
The lengths of three rods are 5cm, 7cm, and 15cm. How can you use these rods to measure a length of 13cm? 
2. Making an organised list 
Example Question 
Five students hold a chess tournament. Each of the students plays each of the other students just once. How many different games are played? 
3. Making a table 
Example Question 
Two dice both have faces numbered from 1 through to 6. Suppose that you role the two dice. What is the probability of rolling a sum of 8 in the uppermost faces? 
4. Solving a simpler related problem 
Example Question 
The houses on Thomas Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7? 
5. Finding a pattern 
Example Question 
What is the sum of the following series of numbers? 
6. Guessing and Checking 
Example Question 
Arrange the counting numbers from 1 to 6 in the circles so that the sum of the numbers along each side of the triangle is 10. 
1.10 Problem Solving Strategies
1. Acting out the problem 
Example Question 
Suppose that you buy a rare stamp for $16, sell it for $22, buy it back for $30, and finally sell it for $35. How much money did you make or lose? 
2. Working backwards 
Example Question 
At the end of a school day, a teacher had 15 crayons left. The teacher remembered giving out 13 of all her crayons in the morning, getting 8 back at recess, and giving out 9 crayons after lunch. How many crayons did the teacher have at the start of the day? 
3. Writing an Equation 
Example Question 
The triple of what number is sixteen greater than the number? 
4. Changing your point of view 
Change your approach 

Are you assuming something thats not in the question 
Example Question 
Draw four continuous line segments through the nine dots 
5. Using Reasoning 
Example Question 
A school has 731 students. Prove that there must be at least 3 students who have the same birthday. 
6. Miscellaneous 
Example Question 
Three apples and two pears cost 78 cents. But two apples and three pears cost 82 cents. What is the total cost of one apple and one pear? 
2.1 Logical Approach to Problem Solving
4 Steps to Problem Solving 
Step 1 
Understand the problem 
Step 2 
Develop a plan 
choose a problem solving strategy 
Step 3 
Carry out the plan 
Step 4 
Reflect 
Mathematical Terms used in the Olympiad 
Standard Form 
1358 
Expanded Form 
1x1000+3x100+5x10+8x1 
Exponential Form 
1x10^{3}+3x10^2+5x10+8x1 
Whole numbers 
0,1,2,3,... 
Counting numbers 
1,2,3,... 
Divisibility 
A is divisible by B, if B divides into A with zero remainder 
If so, B is a factor of A 
Prime number 
counting number greater than 1, which is divisible only by itself and 
Composite number 
counting number greater than 1 which is divisible by a counting number other than 1 and itself 
A number is factored completely 
when it is a product of prime numbers 
Order of Operation 
BODMAS 
common or simple fraction 
a/b where a and b are whole numbers and b is no zero 
unit fraction 
common fraction with a numerator of 1 
proper fraction 
a/b where a < b 
improper fraction 
a/b where a > b 
complex fraction 
numerator or denominator contains a fraction 
20th century 
100 year period 19012000 inclusive 
average of a set of N numbers 
sum of the N numbers divided by N 
acute angle 
less than 90 degrees 
right angle 
90 degrees 
obtuse angle 
greater than 90 degrees 
straight angle 
180 degrees 
reflex angle 
more than 180 degrees and less than 360 degrees 
scalene triangle 
no equal angles 
isosceles triangle 
2 equal angles 
equilateral 
3 equal angles 
rightangled 
90 angle 
congruent shapes 
shapes on the same plane whose sides and angles are the same 
2.2 Types of Problems
1. Translation Problems 
translate word sentences to mathematical sentences 
Example Question 
Farmer Joe bought 2 bags of feed for $4 each and 1 bag of feed for $3. How much did the feed bags cost altogether? 
2. Application Problems 
'realworld' problems, usually involve calculations with money, to find discounts, profits or cost of items 
Example Question 
Shop A is offering a 10% discount on 34cm colour TV sets priced normally at $379. Meanwhile Shop B is offering 15% discount on the same sets priced normally at $409. Which shop should you purchase the TV from? 
3. Process Problems 
Usually require using general problem solving steps and specific strategies. May use shortcuts when aware of patterns 
Example Question 
The first 4 triangular numbers are 1, 3, 6, 10. What will the 10th triangular number be? 
4. Puzzle Problems 
like riddles 
Example Question 
Three Australian students who were born in different countries have last names Brown, Black and Bright. Their first names are Jim, John and Jane but not necessarily in that order. Using the information below can you determine the full name of each student? Brown was born in Australia Bright has never been to Malaysia Jane was born in England Jim was born in Malaysia 

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