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# Mathematics Olympiad Cheat Sheet by peterwongau

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 underneath2. pair up and sum3. sums of pairs are the same4. Since sums are the same, multiple sum by number of pairs5. Divide by 2 Example: Find 2+4+6+...+78+802+4+6+...+78+8080+78+...+6+4+282*40=32803280/2=16402+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 EVENends 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. factorise2. multiply the factors that are commononly those factors that have a pair example HCF of 240 and 924240=2x5x2x2x3x2924=3x2x7x11x2HCF=2x3x2=12 LCM Of all the multiples of the 2 numbers, its the smallest multiple they have in common Lowest Common Multiple method 1. factorise2. multiply the factors that are common and factors they dont have in common example LCM of 120 and 140120=2x2x2x3x5140=2x2x3x7LCM=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 solveFind 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 solve1. 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 solvework 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 coins9x50c=\$4.501x20c=20c3x5c=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 = 14x = 2y = 6 example 7x + y = 18 (1)2x + 2y = 12 (2) (1) x 214x + 2y = 36 (1a)(1a) - (2)12x = 24x = 2y = 4 2. Solving by Substi­tution method 1. rearrange one equation for y2. 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) = 10x = 2y = 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 AustraliaBright has never been to MalaysiaJane was born in EnglandJim was born in Malaysia