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Mathematics Olympiad Cheat Sheet by

Cheatsheet for to help prepare for the Mathematics Olympiad

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
=75-2+75-1+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=N2
eg 2x2=22=4
Cubed Numbers
NxNxN=N3
eg 2x2x2=23=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 = a3 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 old-style 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
1x103+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 1901-2000 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
right-­angled
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
'real-­world' 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 short-cuts 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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