1

Unit 1: Arithmetic

1.1 Proportion

Demonstrate understanding of primary ideas of proportion

Solve problems involving direct and inverse proportions (with 2 or more variables) A

typical 3 variable problem: Find the number of people required to complete a certain

number of jobs in a certain number of days e.g. (J04 Q17)

1. J95 Q8

What is the smaller angle between the minute and hour hands of a 12-hour clock at 3.40 pm?

(A) 150 (B) 160 (C) 130 (D) 120 (E) 180

1.2 Ratio

Demonstrate understanding of primary ideas of ratio

1. J97 Q3

In the diagram, calculate the ratio of the area of the shaded region to the total area of the two

identical smaller circles.

(A) 1 : 1 (B) 1 : 2 (C) 1 : 3 (D) 2 : 3 (E) 3 : 4

2. J97 Q25

Three boys, Tom, John and Ken, agreed to share some marbles in the ratio of 9 : 8 : 7

respectively. John then suggested that they should share the marbles in the ratio 8 : 7 : 6

instead. Who would then get more marbles than before and who would get less than before if

the ratio was changed?

2

3. J99 Q6

The diagram below shows three semi-circles whose centers all lie on the same straight line

ABC. Suppose BCAB 2 .

The ratio of the shaded area to the area of the largest semi-circle is

(A) 1:2 (B) 4:9 (C) 1:3 (D) 2:3 (E) 2:5

1.3 Rates

Demonstrate understanding of primary ideas of rate

Use the formula – time

distance speed

Use the formula – timetotal

distance total speed average

Convert units (e.g. km/h to m/s and vice versa)

1. J95 Q25

Two pipes can be used to fill a swimming pool. The first can fill the pool in three hours, and

the second can fill the pool in four hours. There is also a drain that can empty the pool in six

hours. Both pipes were being used to fill the pool. After an hour, a careless maintenance man

accidentally opened the drain. How long more will it take for the pool to fill?

2. J96 Q4

Town A and Town B are linked by a straight road. A factory is sited along the road such that

it is twice as far away from Town A as its distance from Town B. A truck left Town B at 9.00

am and reached the factory 1 hour later. A car which travels three times faster than the truck

need to reach the factory at the same time as the truck. What time must the car leave Town A?

(A) 8.20 am (B) 8.40 am (C) 9.20 am (D) 9.40 am

(E) 10.00 am

3. J99 Q14

If John walks from home to school at the speed of 4 km per hour, and walks back at the speed

of 3.5 km per hour, find the average speed in km per hour for the whole trip.

(A) 3.75 (B) 3.6 (C) 15113 (D)

543 (E)

323

A B C

3

1.4 Percentage

Calculate percentage (including percentage increase and decrease)

1. J96 Q1

If the side of a square is increased by 30%, the area of the square will increase by

(A) 30% (B) 60% (C) 69% (D) 900% (E) None of the above

2. J97 Q7 (percentage)

Originally 32 of the students in a class failed in an examination. After taking a re-examination,

40% of the failed students passed. What is the total pass percentage of the class?

(A) %2632 (B) %33

31 (C) %40 (D) %60 (E) %73

31

3. J97 Q8 (percentage)

A company’s sales increase by 20% in 1993 followed by another 25% in 1994. The sales

decreased by 25% in 1995, however. This was followed by another decrease of 20% in 1996.

By what percentage did the company’s sales increase or decrease over this four-year period?

(A) Increased by 5% (B) Decreased by 5% (C) Increased by 10%

(D) Decreased by 10% (E) No increase or decrease

1.5 Statistics

Use the formulae – arithmetic mean = n

xxx n 21

1. J96 Q21 (arithmetic, average, inequality, reasoning)

Class A, with 15 students in the class, scored an average of 94 marks in a mathematics test.

The maximum possible score of the test is 100 marks. What is the lowest possible score that

any of the 15 students could have scored?

1.6 Relative velocity

Use primary ideas of relative velocity to solve problems

PV : True (actual) velocity of a moving object P relative to the Earth.

PV : True speed (ground speed) of P

QPV / : Relative (apparent) velocity of a moving object P relative to a moving object Q

(observer)

PQV / : Relative (apparent) velocity of a moving object Q relative to a moving object P

(observer)

Relative velocity equation

4

PV = QPV / + QV or QPV / = PV + (– QV )

1. J95 Q16

Two trains are each traveling towards each other at 180 km/h. A passenger in one train

notices that it takes 5 seconds for the other train to pass him. How long is the second train?

(A) 100 m (B) 200 m (C) 250 m (D) 400 m (E) 500 m

2. J95 Q17

In a river with a steady current, it takes Bionic Woman 6 minutes to swim a certain distance

upstream, but it takes her only 3 minutes to swim back. How many minutes would it take a

doll of the Bionic Woman to float this same distance downstream?

(A) 8 minutes (B) 9 minutes (C) 10 minutes (D) 11 minutes

(E) 12 minutes

3. J99 Q30

Two men are walking at different steady paces upstream along the bank of a river. A ship

moving downstream at constant speed takes 15 seconds to pass the first man. Five minutes

later it reaches the second man and takes 10 seconds to pass him. Starting then, how long will

it take for the two men to meet? (Give you answer in terms of seconds).

1.7 Binary Operation

Perform binary operation

A binary operator in mathematics is defined as an operator defined on a set that takes two

elements of the set and returns a single element. An example would be integer multiplication

"" where a, b are both integers and ab returns an integer.

1. J97 Q9

The operation is defined by: a b = a2 – b

2.

Evaluate (1997 1996) (1996 1995).

(A) 3991 (B) 3993 (C) 7984 (D) 15968 (E) None of the above

2. J99 Q16

Let be the binary operator on positive integers defined by a b = ab.

Consider the following identities:

(i) a b = a b

(ii) (a b) c = a (b c)

(iii) a (b + c) = (a b)+(a c)

5

(iv) (a + b) c = (a c)+(b c)

(A) All are true (B) (ii) and (iii) are true (C) (iii) and (iv) are true

(D) (iii) is true (E) None is true.

Unit 2: Mathematical Reasoning

2.1 Logic and paradoxes

Use strategies of making suppositions, eliminating possibilities and making logical

deductions to evaluate the truth of statements

1. J00 Q20

Four people, A, B, C and D are accused in a trial. It is known that

if A is guilty, then B is guilty;

if B is guilty, then C is guilty or A is not guilty;

if D is guilty, then A is guilty and C is not guilty;

if D is guilty, then A is guilty.

How many of the accused are guilty?

(A) 1 (B) 2 (C) 3 (D) 4

(E) Insufficient information to determine

6

Unit 3: Algebra

3.1 Algebraic representation and formulae

Use letters to express generalized numbers and express arithmetic processes

algebraically

Use the strategy: Students should note some questions need not be solved

algebraically. They can consider specific cases by substituting appropriate numbers,

reducing the problem to an arithmetic one

3.2 Algebraic Manipulation

Manipulate/Simplify algebraic expressions (including algebraic fractions). (Students

should be able to use tricks like adding “new” terms while still maintaining integrity

of question to solve problems e.g. J98 Q26)

(Partial fractions) Express an (algebraic) fraction as a difference of two fractions

(Classic example: 1

11

)1(

1

nnnn)

Manipulate algebraic fractions/expressions in an equation (usually to substitute the

result into another expression/equation to solve a given problem e.g. J04 Q12)

3.3 Algebraic Manipulation (Expansion and Factorisation)

Factorise expressions of the form ayax

Know and use the identity – ))((22 yxyxyx and other equivalent forms e.g.

yx

yxyx

, yxyxyx ))(( …

Know and use the identity - 222 2)( yxyxyx and other equivalent forms e.g.

xyyxyx 4)()( 22 , 2

2

2 21

)1

( xx

xx

… (e.g. J04 Q23, J01 Q21) (Note:

to solve some questions, repeated use of this identity is necessary (e.g. J04 23, J01

Q21))

Factorise trinomials

Factorisation by grouping (students should be comfortable with atypical scenarios

involving more than 4 terms e.g. J03 Q18 involves factorization of

1222 234 nnnn )

Know and use the identity – ))(( 2233 yxyxyxyx

Know and use the technique of completing the square (e.g. J00 Q9: to determine the

minimum or maximum value of expression)

Expansion and Factorisation

Some useful identities

7

))((

))((

33)(

33)(

))((

2)(

2)(

2233

2233

32233

32233

22

222

222

babababa

babababa

babbaaba

babbaaba

bababa

bababa

bababa

The absolute value of function (e.g. square root of a square)

The absolute value (or modulus) of x means the numerical value of x, not considering its sign,

and is denoted by a .

Is aa 2 for all real number a?

Consider:

When a = 2, aa 24222

When a = –2, aa 24)2( 22

0 if

0 if 2

aa

aaa (or aa 2 )

1. J95 Q6

The sum of two positive numbers equals the sum of the reciprocals of the same two numbers.

What is the product of these two numbers?

(A) 1 (B) 2 (C) 4 (D) 21 (E)

41

2. J95 Q24

If x and y satisfy 722 yx , find the maximum value of 422 22 xyx .

3. J96 Q15

If the value of 76x – 19y is 114, the value of 36x – 9y is

(A) 54 (B) 60 (C) 88 (D) 92 (E) 108

8

4. J96 Q16

Let a < 0. Find 22 )1( aa in terms of a.

(A) 1 (B) –1 (C) 2a – 1 (D) 1 – 2a (E) None of the above

5. J96 Q18

If 51

xx , find the value of

xx

1 is________.

6. J97 Q2

Given that 19621007100610051998199719961995 2323 xxxxxx , the value of

123 xxx _______________.

(A) –4 (B) 3 (C) –3 (D) 1 (E) –1

7. J97 Q14

Three boys agree to divide a bag of marbles in the following manner. The first boy takes one

more than half the marbles. The second takes a third of the number remaining. The third boy

finds that he is left with twice as many marbles as the second boy. The original number of

marbles

(A) is 8 or 38 (B) cannot be determined from the given data

(C) is 20 or 26 (D) is 14 or 32 (E) is none of these

8. J97 Q15

Coffee A and coffee B are mixed in the ratio x : y by weight. A costs $50/kg and B costs

$40/kg. If the cost of A is increased by 10% while that of B is decreased by 15%, the cost of

the mixture per kg remains unchanged. Find x : y.

(A) 2 : 3 (B) 5 : 6 (C) 6 : 5 (D) 3 : 2 (E) 55 : 34

9. J98 Q26

Let m and n be two integers such that 698 mnnm . Find the largest possible value of m.

10. J98 Q27

Find the largest value of x which satisfies the equation 222222 )73()2()2)(54()54( xxxxxxxx .

9

11. J99 Q2

If we increase the length and the width of a rectangle by 10 cm each, the area of the rectangle

will increase by 300 cm2. The perimeter of the original rectangle in cm is

(A) 28 (B) 30 (C) 36 (D) 40 (E) 50

12. J99 Q9

Given that 11142 xx , the value of 12 xx is

(A) 71 (B) 81 (C) 91 (D) 47 (E) 63

13. J99 Q26

Let 4

111

yx. What is the value of

xyxy

xxyy

2

232

?

14. J00 Q9

For any real numbers a, b and c, find the smallest possible value the following expression can

take:

23730185273 222 cabcba .

(A) 190 (B) 192 (C) 200 (D) 237 (E) 239

15. J00 Q13

In the following diagram, ABCD is a rectangle and ADEF, CDHG, BCLM and ABNO are

square. Suppose the perimeter of ABCD is 16 cm and the total area of the four squares is 68

cm2. Find the area of ABCD in cm

2.

N

A

O H

CB

M

F

G

L

E

D

(A) 15 (B) 20 (C) 25 (D) 30 (E) 40

10

3.4 Solutions of Equations

Construct equations from given situations

Solve linear equations in one unknown

Manipulate and/or solve simultaneous equations (Students should be comfortable with

atypical questions e.g. 2 equations but many unknown (J04 Q1). Such questions can

usually be simplified further through cancellation of “excess variables” (J04 Q1),

clever manipulation (J04 Q27)

Solve quadratic equations by factorization

Solve quadratic equations by completing the square

Solve quadratic equations by the use of formula

Solve complex equations (e.g. surds, polynomials of higher degrees) through non-

routine methods (e.g. by using a suitable substitution to simplify equation (e.g. from

polynomial of high degree to trigonometric function S01 Q25, from surds to quadratic

J98 Q15, from exponential to quadratic J96 Q11)

1. J95 Q13

When some people sat down to lunch, they found there was one person too many for each to

sit at a separate table, so they sat two to a table and one table was left free. How many tables

were there?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

2. J95 Q21

John is now twice as old as Peter. If their combined age is 54 years, what is their combined

age when Peter is as old as John is now?

3. J95 Q30

On a plane, two men had a total of 135 kilograms of luggage. The first paid $12 for his

excess luggage and the second paid $24 for his excess luggage. Had all the luggage belonged

to one person, the excess luggage charge would have been $72. At most how many kilograms

of luggage is each person permitted to bring on the plane free of additional charge?

4. J96 Q11

If x and y satisfy the following simultaneous equations 645121616 yx and 22444 yx ,

find the sum yx .

(A) 5.5 (B) 6.5 (C) 8.5 (D) 12.5 (E) 16.5

11

5. J98 Q1

A bag contains 28 marbles which are coloured either red, white, blue or green. There are 4

more red marbles than white ones, 3 more white marbles than blue ones and 2 more blue

marbles than green ones. Find the number of white marbles.

6. J98 Q15

It is given that a, b are two positive numbers satisfying

)5(3)( babbaa .

Find the value of baba

baba

223.

7. J99 Q4

Let x denote the absolute value of a number x defined by

x if 0x

x

–x x < 0.

The number of solutions of the equation 221 x is

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

8. J99 Q24

When I am as old as my father is now, my son will be seven years older than I am now. At

present, the sum of the ages of my father, my son and myself is 100. How old am I?

9. J00 Q16

A student has taken n examinations and 1 more examination is upcoming. If he scores 100 in

the upcoming examination, his overall average (of the n + 1 examinations) will be 90; and if

he scores 60 in the upcoming examination, his overall average will be 85. Find the number n.

(A) 5 (B) 6 (C) 7 (D) 8 (E) 9

12

3.5 Functions (Absolute Function)

Know and use the properties of absolute function

The absolute value of function (e.g. square root of a square)

The absolute value (or modulus) of x means the numerical value of x, not considering its sign,

and is denoted by x .

0 if

0 if

xx

xxx

3.6 Roots

Use the formulae for the product and sum of roots

Use the condition for a quadratic equation to have two real roots, two equal roots and

no real roots

Determine the existence of integral/rational roots for quadratic equations through

computation of the discriminant

Sum of roots and Product of roots

If and are two roots of the quadratic equation 02 cbxax , then

+ = a

b

= a

c

1. J99 Q29

Let a and b the two real roots of the quadratic equation

043)1( 22 kkxkx

where k is some real number. What is the largest possible value of 22 ba ?

2. J00 Q24

Suppose the equation

01)4(2 axax

has two solutions which differ by 5. Find all possible values of a.

13

3.7 Indices

Apply the laws of indices

Perform operations with indices

Determine the nth root of a number

Solve indicial equations, solve equations of the form ba x

1. J95 Q1

The simplest expression for 20

40

4

2 is

(A) 1 (B) 4 (C)

20

2

1

(D) 202 (E) 182

2. J95 Q7

What is the value of x which satisfies

19952 19952 19952 19952 19952 19952 19952 19952 = x2 ?

(A) 1996 (B) 1997 (C) 1998 (D) 1999 (E) 2000

3. J96 Q9

Suppose 19961996 19961996 19961996 = x1996 , what is the value of x?

1996 terms

(A) 1996 (B) 1997 (C) 1998 (D) 1999 (E) 2000

4. J96 Q10

Solve the equation 27

2)1()1()1( xxx .

(A) 5 (B) 6 (C) 8 (D) 12 (E) 15

5. J97 Q1

Given that 19981998 19971998 = 19971998x , find the value of x.

(A) 0 (B) 1 (C) 1996 (D) 1997 (E) 1998

14

6. J97 Q11

If 13574a , 3575b and 23572c , find the sum of all the digits in c

ab.

(A) 1 (B) 10 (C) 15 (D) 357 (E) None of the above

7. J00 Q2

The fifth root of 555 is

(A) 55 (B) )15( 5

5 (C) 545 (D)

455 (E) 55

5

8. J00 Q19

How many integer solutions does the following equation have?

1120002

xxx .

(A) 1 (B) 3 (C) 3 (D) 4 (E) 5

3.8 Standard Form

Use the standard form nA 10 where n is a positive of negative integer, and 101 A

Deduce the number of digits of a number from its standard form (or its variations e.g.

standard form minus one (J04 Q29)) (Questions like this typically require students to

extract a2 and a5 from a given number, this allows the number to be expressed in the

standard form (J02 Q7, J98 Q2)

Write algebraic expressions (e.g. abcd) as as linear combinations of ,10,10,10 210

1. J95 Q2

If 077119823.521047.8 3 , what is 38047.0 equal to?

(A) 0.521077119823 (B) 52.1077119823 (C) 521077.119823

(D) 0.00521077119823 (E) 0.052107119823

2. J96 Q7

The square of the number 12345678 is an m-digit number. What is the value of m?

(A) 13 (B) 14 (C) 15 (D) 16 (E) 17

15

3. J98 Q2

What is the number of digits in the number 86 54 ?

3.9 Applications of algebra in arithmetic computation

Simplify arithmetic computation by

pairing terms in numerical expressions (e.g. pairing 1st and last term, pairing

adjacent terms)

using the method of difference (such a method typically involves the

knowledge and use of partial fractions)

using a suitable algebraic form to model the question

factorizing (repeatedly e.g. J99 Q15, J98 Q3) complex numerical expressions

using a suitable substitution

using the technique – cancellation of numerators and denominators (by first

writing numerical expressions into suitable fractional forms)

using estimation and approximation

expressing numbers as linear combinations of ,10,10,10 210

expressing each numbers as a difference of two numbers Classic

case: (J02 20 – 9 + 99 + 999 + … = 10 – 1 + 100 – 1 + 1000 – 1 +…)

extrapolating the result of simple arithmetical calculation to cases involving

large numbers (e.g. J04 Q8 – What is the sum of all the digits in the number

2004102004 ?)

1. J95 Q22

Evaluate 123458123456123457

2469122

.

2. J96 Q14

Evaluate 180018

)1897645)(1897645()1987654)(1987654( .

3. J96 Q25

Evaluate )1()1()1)(1)(1(101

11

4

1

3

1

2

1 n

.

4. J96 Q28

What is the unit digit for the sum 3333333 19654321 ?

5. J97 Q4

Which of the following is the closest value to

)05.0)(367,19(

)000,487)(001,621,9()300,027,12)(000,487( ?

(A) 10,000,000 (B) 100,000,000 (C) 1,000,000,000

16

(D) 10,000,000,000 (E) 100,000,000,000

6. J97 Q12

The difference between the sum of the last 1997 even natural numbers less than 4000 and the

sum of the last 1997 odd numbers less than 4000 is

(A) 1996 (B) 1997 (C) 1998 (D) 3994 (E) 3996

7. J98 Q3

Find the value of

88

44222

248252

)248252()248252(1000

.

8. J98 Q8

Find the value of 222222 199819974321 .

9. J98 Q10

Find the value of

199819971997199719971998 .

10. J99 Q21

What is the product of

20002000

11)

1002

11(

1001

111001

222

?

11. J00 Q1

Let x be the sum of the following 2000 numbers:

44444 , 44, ,4 .

Then the last four digits (thousands, hundreds, tens, units) of x are

(A) 0220 (B) 0716 (C) 1884 (D) 2880 (E) 5160

12. J00 Q4

Find the value of the product )2000

11)(

1999

11()

3

11)(

2

11(

2222 .

(A) 4000

2001 (B)

2000

1001 (C)

201

101 (D)

40

21 (E)

20

11

2000 digits

17

13. J00 Q7

Let nS n

n

1)1(4321 . Find the value of 20012000 SS .

(A) –1 (B) 0 (C) 1 (D) 2 (E) 3

14. J00 Q14

Evaluate

)99

98

4

3

3

2)(

100

99

4

3

3

2

2

1()

99

98

4

3

3

2

2

1)(

100

99

4

3

3

2( .

15. J00 Q28

Evaluate 2222211111 .

2000 digits 1000 digits

(Hint: Your answer should be an integer.)

3.10 Sequences and Series

Continue/complete given number/alphabetical sequences

Apply the first principles of arithmetic progressions (pairing of terms i.e. first and last

etc)

Recognise arithmetic progressions

Use the formula for the nth term to solve problems involving arithmetic progressions

Use the formula for the sum of the first n terms to solve problems involving

arithmetic progressions

Use the result 1 nnn SST

Recognise geometric progressions

Use the formulae for the sum of the first n terms to solve problems involving

geometric progressions

Determine the largest (smallest) term of a sequence by comparing the nth and (n+1)th

term

Use the method of differences to obtain the sum of a finite series e.g. by expressing

the term in partial fractions

Arithmetic Progression (AP)

The nth term of an AP (with common difference d) is given by

dnaan )1(1

The sum of the first n terms of an AP (with common difference d) is given by

))1(2(2

)(2

1121

1

1 dnan

aan

aaaa nn

n

i

18

If x, y and z are three consecutive terms of an AP, then

zyyz or zxy 2 or 2

zxy

(arithmetic mean)

Geometric Progression (GP)

The nth term of a GP (with common ratio r) is given by 1

1

n

n raa

The sum of the first n terms of a GP (with common difference r) is given by

r

ra

r

raaaaa

nn

n

n

i

1

)1(

)1(

)1( 1121

1

1

If x, y and z are three consecutive terms of a GP, then

x

y

y

z or xzy 2 or xzy (geometric mean)

Some important formulae

The sum of first n natural numbers:2

)1(4321

nnn

The sum of first n odd numbers: 2

2

)121()12(7531 n

nnn

The factorization of 1nx : )1)(1(1 221 xxxxxx nnn

(e.g. )1)(1(12 xxx )

1. J96 Q17

The value of 1100332211 is .

2. J97 Q17

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

etc.

Pascal’s triangle is an array of positive integers (see above), in which the first row is 1, the

second row is two 1’s, each row begins and ends with 1, and the kth integer in any row when

it is not 1, is the sum of the kth and (k–1)th numbers in the immediately proceeding row. Find

the ratio of the number of integers in the first n rows which are not 1’s and the number of 1’s.

(A) 12

2

n

nn (B)

24

2

n

nn (C)

12

22

n

nn (D)

24

232

n

nn

(E) None of the above.

3. J97 Q18

19

The integers greater than one are arranged in five columns as follows:

A B C D E

2 3 4 5

9 8 7 6

10 11 12 13

17 16 15 14

In which columns will the number 1000 fall?

(A) A (B) B (C) C (D) D (E) E

4. J97 Q21

In a game, a basket and 16 potatoes are placed in line at equal interval of 3 m. (Note that the

basket is placed at one end of the line). How long will a competitor take to bring the potatoes

one by one into the basket, if he starts from the basket and run at an average speed of 6 m a

second?

5. J98 Q24

The sequence },,,{ 321 aaa is defined by ,21 a and naa nn 21 for ,3,2,1n . Find

the value 100a .

6. J99 Q1

The next letter in the following sequence

B, C, D, G, J, O, ____

is

(A) P (B) Q (C) R (D) S (E) T

7. J99 Q11

The number 1001997 is expressed as a sum of 999 consecutive odd positive integers. The

largest possible such odd integer is

(A) 1997 (B) 1999 (C) 2001 (D) 2003 (E) 2005

8. J99 Q25

Let n! denote the product 12)2()1( nnn . For what value of the positive integer

n is !/3

100n

n

largest?

20

9. J00 Q5

Consider the following array of numbers:

A B C D E

2 5 8

23 20 17 14 11

26 29 32

47 44 41 38 35

In which column does the number 2000 appear?

(A) A (B) B (C) C (D) D (E) E.

10. J00 Q6

Find the sum of all positive integers which are less than or equal to 200 and not divisible by 3

or 5.

(A) 9367 (B) 9637 (C) 10732 (D) 12307 (E) 17302.

11. J00 Q10

For which positive integer k does the expression k

k

001.1

2

have the largest value?

(A) 1998 (B) 1999 (C) 2000 (D) 2001 (E) 2002

21

3.11 Inequalities

Know and use the properties of inequalities

(students also need to be aware of self-evident properties of real numbers; refer to

classic: J99 Q10)

e.g. naa n for 1 ,1 If (J04 Q9)

e.g. 10

ba

a for 0, ba (J02 Q15)

Manipulate inequalities

Substitute equations into inequalities

Construct inequalities from given situation

Solve linear inequalities

Solve quadratic inequalities (by factorization, etc)

Solve quadratic inequalities through non-routine techniques (using the property of

integers)

Solve cubic inequalities

Students should be able to solve these inequalities using non-routine methods e.g. by

approximation which in terms requires familiarity with the values of manageable”

numbers raised to the power of n, where n is a small integer J98 Q5

Solve complex inequalities (involving combination of different functions) e.g.

(exponential and linear), involving surds J98 Q22 by non-routine methods (e.g. trial

and error) (e.g. manipulating inequalities J98 Q22), (involving greatest integer

functions by trial and error J96 Q22)

Solve simultaneous inequalities/equations/inequations (e.g. S03 Q25)

Use “Squeeze” theorem i.e. find suitable lower and upper bounds for

algebraic/numerical expressions

Compare the magnitude of numbers using a variety of techniques e.g. J03 by

evaluating their difference, J02 rewriting numbers to that they have the same

exponent, J01 rewriting numbers as fractions with the same numerator but different

denominator using the identity yx

yxyx

, J96 by evaluating their

differences/ considering specific cases

Determine the intersection of solution sets of at least 2 inequalities

1. J95 Q3

If x is a positive number, which of the following expressions must be less than 1?

(A) x

1 (B)

x

x1 (C) 2x (D)

x

x1 (E)

1x

x

22

2. J95 Q15

If 10 x , xxy and yxz , what are the three numbers arranged in order of increasing

magnitude?

(A) x, y, z (B) x, z, y (C) y, z, x (D) z, y, x (E) z, x, y

3. J96 Q26

If a, b, c and d are positive integers such thatd

c

b

a1 , arrange the following quantities in

ascending order.

1 ,,,,ca

db

ac

bd

c

d

a

b

.

4. J96 Q27

Find all possible real values y such that 16847 yx and 12135 yx .

5. J97 Q28

If the solution of the inequality 062 axx is 6 xc , find c.

6. J98 Q5

Find the positive integer n such that 18000060000 3 n and the unit digit of 3n is 3.

7. J98 Q22

What is the smallest positive integer n such that 02.03414 nn ?

8. J99 Q3

Suppose 26 a and 622 b . Then

(A) ba (B) ba (C) ba (D) ab 2 (E) ba 2

9. J99 Q7

Let 482a , 363b , and 245c . Then

(A) cba (B) abc (C) acb (D) cab (E) bca .

23

10. J99 Q8 (properties of fraction)

The integer part of the fraction

19991

19851

19841

1

is

(A) 121 (B) 122 (C) 123 (D) 124 (E) 125

11. J99 Q10

Let a and b be two numbers such that a > b. Consider the following inequalities:

(i) 22 ba (ii) ba11 (iii) 1

ba (iv) 0ab

(A) All are true (B) only (i) is true (C only (ii) is true

(D) (i), (ii), (iii) are true (E) None is true

3.12 Surds

Perform operations with surds, including rationalizing the denominator

3.13 Logarithm

Use the laws of logarithm

3.14 Identities

Substitute appropriate values for x into identities by observation (usually to find

solutions of (linear combinations of) coefficients)

Identities

)()( xQxP )()( xQxP for all values of x

To find unknowns in an identity,

(a) substitute values of x, or

(b) equate coefficients of like powers of x.

1. J00 Q17

If 200

200

2

210

1002 2)733( xaxaxaaxx , find the value of

20019886420 2222222 aaaaaaa .

(A) 0 (B) 1 (C) 200 (D) 2000 (E) 1007

24

S R

P Q

3.15 Binomial theorem

Use the Binomial Theorem for expansion of nba )( for positive integral n

The Binomial Theorem for positive integer, n

nba )( = na + baC nn 1

1

+ 22

2 baC nn + 33

3 baC nn + … + nb

There are n + 1 terms. The powers of a are in descending order while the powers of b are in

ascending order. The sum of the powers of a and b in each term of the expansion is always

equal to n.

1rT = rrn

r

n baC

If a = 1, nb)1( = 1 + bCn

1 + 2

2bCn + 3

3bCn + … + nb

1rT = r

r

n bC

Unit 4: Geometry

4.1 Mensuration: Perimeter, Area and Volume

Calculate area and perimeter of geometrical figures (including triangles, circles,

sectors, squares, rectangles etc)

Calculate area of “irregular”/geometrical figures indirectly

Know and use the formulae for surface area and volume of spheres, cubes, cones

1. J95 Q20

An equilateral triangle ABC has an area of 3 and side of length 2. Point P is an arbitrary

point in the interior of the triangle. What is the sum of the distances from P to AB, AC and BC?

2. J95 Q26

In the diagram, congruent radii PS and QR intersect tangent SR. If the two disjoint shaded

regions have equal areas and if PS = 10 cm, what is the area of quadrilateral PQRS?

25

4.2 Radian measure

Solve problems including arc length and sector of a circle, including knowledge and

use of radian measure

4.3 Angles

Use the following geometrical properties alternate angles

sum of angles at a point

exterior angle = sum of interior opp angles

angle sum of triangle

1. J96 Q8

In the diagram, ABCD is a rectangle with AD = 2AB. M and N are midpoints of AD and BC

respectively. Triangle ABE is an equilateral triangle. Calculate MEN.

A

CB

DM

N

E

2. J96 Q29

In the following figure, AB = AC = BD. Find y in terms of x.

B

A

CD

xy

26

4.4 Properties of Geometrical Figures

Know the properties of (equilateral and isosceles) triangles

Know the properties of quadrilaterals (square, rhombus, parallelogram, rectangle,

kite)

1. J95 Q9

A four-sided closed figure has opposite sides equal in length. Which of the following

statements about this figure must be true?

(A) If all its sides are equal in length, then the diagonals are equal in length.

(B) If the adjacent sides are perpendicular, then all its sides are equal in length.

(C) If its diagonals are equal in length, then the adjacent sides are perpendicular.

(D) If its diagonals are perpendicular, then the adjacent sides are perpendicular.

(E) If its diagonals are equal in length, then all its sides are equal in length.

2. J96 Q20

ABCD is a trapezium with AB parallel to DC. The point E on CD is such that DAE =

BAE and CBE = ABE. Given that AD = 13 cm and BC = 12 cm, calculate the length of

CD.

4.5 Polygons

Calculate interior and exterior angles of polygons

1. J99 Q5

The sum of the angles

A + B + C + D + E + F + G

in the diagram is

(A) 240 (B) 280 (C) 350 (D) 360 (E) 420

G

F

ED

C

B

A

27

4.6 Three Dimensional Figures

Draw the nets for a given solid (cube etc)

Know the relationship between a cone and a sector

Cones

A cone is a solid defined by a closed plane curve (forming the base) and a point (not on the

same plane) called the vertex. When the base of a cone is a circle, it is called a circular cone.

A right circular cone can be generated by the rotation of the right-angled triangle VOC about

VO, which represents the height of the cone. Every point on the circumference of the base is

the same distance l from the vertex V. The length l is called the slant height.

Answer the following questions before you proceed to deduce the formula for the curved

surface of a cone:

1. If a cut is made along VC and the cone is opened up and laid flat, what does it form?

__________________________________

2. What length of the sector corresponds to the slant height l?

__________________________

3. What length in the cone corresponds to the arc C1C2 in the sector?

____________________

l

l l

h

r

V

C

O

V

V

C1

C1

C2

C2

Cut along VC

28

Now, fill in the blanks:

Area of sector Arc length

Area of circle Circumference 360

down an expression for

1. J99 Q27

The following diagram shows a solid cube of volume 1 cm3. Let M be the midpoint of the

edge BC. What is the shortest distance in cm for an ant crawl from the vertex A to M?

E

H

F

A

G

C

M

B

D

Given that

360nceCircumfere

length Arc , write down the ratio of

360

in terms of r and l.

circle of AreanceCircumfere

length Arc sector of Area

Curved surface area of a cone = ____________________

Total surface area of a closed cone = _________ + _________

29

4.7 Circle Properties

Solve problems using the geometrical properties:

a straight line drawn from the centre of a circle to bisect a chord (not a

diameter), is perpendicular to the chord and vice versa

rt. angle in a semi-circle

angle at centre is twice angle at circumference

angles in the same segment

angles subtended by arcs of equal length

tangent perpendicular to radius

tangents from exterior point are equal

Symmetrical/Angle Properties of Circles

1. a straight line drawn from the centre of a circle bisect a

chord is perpendicular to the chord

2. equal chords are equidistant from the centre of a circle

1. Tangent perpendicular to radius

2. If TA and TB are tangents to a circle with centre O, then

- TA = TB

- ATO = BTO

- AOT = BOT

Angle at centre is twice angle at circumference

Angles in the same segment are equal

Angles at the circumference subtended by equal arcs are equal

Right angle in a semicircle

1. opposite angles of cyclic quadrilateral

2. exterior angle of cyclic quadrilateral

30

Alternate segment theorem

The angle between a tangent and a chord is equal to the angle made by that chord in the

alternate segment.

1. J97 Q13

In the diagram, AM = MB = MC = 5 and BC = 6. Find the area of triangle ABC.

A

C

BM

2. J98 Q14

In the figure below, A, B, C, D are four points on a circle, and the line segments BA and CD

are extended to meet at the point E. Suppose E = 42, and the arcs AB, BC and CD all have

equal lengths. Find the measure of BAC + ACD in degrees.

B

C

E

A

D

31

3. J00 Q26

In the diagram below, A, B, C, D lie on the line segment OE, and AC and CE are diameters of

the circles centred at B and D respectively. The line OF is tangent to the circle centred at D

with the point of contact F. If OA = 10, AC = 26 and CE = 20, find the length of the chord

GH.

O A B DC

G

HF

E

4.8 Loci

Use the following loci and method of intersecting loci sets of points in two dimensions which are equidistant from two given

intersecting straight lines

1. J97 Q16

Line l2 intersects l1 and line l3 is parallel to l1. The three lines are distinct and lie in a plane.

Determine the number of points that are equidistant from all three lines.

4.9 Triangles

Use properties of congruency

Know and use appropriate tests to verify if 2 triangles (figures) are congruent

Use properties of similar figures (including non-triangles)

Know and use appropriate tests to verify if 2 triangles (figures) are similar

Use the relationship between volumes of similar solids

Use the theorem – ratio of area of triangles with common height = ratio of bases

32

Congruent Triangles

Two triangles are congruent if they are identical, i.e. they are of the same shape and size.

ABC is congruent to XYZ (written as ABC XYZ) if AB = XY, BC = YZ,

CA = ZX and A = X, B = Y, C = Z

Test for congruency

SSS: 3 sides on one triangle are equal to 3 sides on the other triangle

SAS: 2 pairs of sides and the included angles are equal

AAS (or ASA): 2 pairs of angles and a pair of corresponding sides are equal

RHS: Right-angled triangle with the hypotenuse equal and one other pair of

sides equal

Similar Triangles

Two triangles are similar if they have the same shape, i.e. the corresponding angles are equal

and the corresponding sides are proportional.

ABC is similar to XYZ (written as ABC XYZ) if ZX

CA

YZ

BC

XY

AB and A =

X, B = Y, C = Z

Tests for similarity

The corresponding angles are equal (AA)

The corresponding sides are proportional

Two corresponding sides are proportional with included angles equal

A

B C

X

Y Z

A

B C

X

Y Z

33

Properties of Similar Figures

If X and Y are two similar solids/figures, then

y

x

y

x

h

h

l

l

22

y

x

y

x

y

x

h

h

l

l

A

A

33

y

x

y

x

y

x

h

h

l

l

V

V

33

y

x

y

x

y

x

h

h

l

l

m

m (if they have the same density)

Triangles sharing the same height

Consider two triangles, with areas 1A and 2A sharing the same height, h.

2

1

2

1

2

1

2

12

1

b

b

hb

hb

A

A

1. J95 Q14

In the diagram, ABC and CDE are right angles. Given that CD = 6 cm, AD = 7 cm and AB

= 5 cm, what is the area of quadrilateral ABED?

A

CB E

D

2. J96 Q2

In the following triangle ABC, M and N are points on AB and AC respectively such that AM :

MB = 1 : 3 and AN : NC = 3 : 5.

1b 2b

h

1A 2A

34

1

B

3

N

C

A

M

3 5

Find the ratio of the area of triangle MNC : area of triangle ABC.

3. J96 Q13

A quadrilateral has sides of length 4 cm, 6 cm, 8 cm and 9 cm respectively. Another similar

quadrilateral has a side of length 12 cm. What is the largest possible perimeter of this similar

quadrilateral?

4. J97 Q6

In the diagram, the radii of the sectors OPQ and ORS are 5 cm and 2 cm respectively. Find

the ratio of the area of the shaded region to the area of the sector OPQ.

5. J98 Q17

In the figure below BAC = 90 and DEFG is a square. If the length of BC is 6

185 and the

area of ABC is 1369, find the area of the square DEFG.

A

B C

GD

FE

6. J98 Q23

In the figure below, AP is the bisector of BAC, and BP is perpendicular to AP. Also, K is the

midpoint of BC. Suppose that AB = 8 cm and AC = 26 cm. Find the length of PK in cm.

O

P Q

S R

35

A

B C

P

K

7. J98 Q25

In the diagram below, ABC is a right-angled triangle with B = 90. Suppose that

2CQ

AQ

CP

BP and AC is parallel to RP. If the area of triangle BSP is 4 square units, find the

area of triangle ABC in square units.

A

BC

R Q

S

P

8 J99 Q12

In the diagram below, ABCD is a square and

n

m

HA

DH

GD

CG

FC

BF

EB

AE .

9. J99 Q17

In triangle ABC, D, E and F are points on the sides BC, AC an AB respectively such that BC =

4CD, AC = 5AE and AB = 6BF. Suppose the area of ABC is 120 cm2, what is the area of DEF

in cm2?

A H

36

4.10 Coordinate Geometry

Calculate the gradient of a straight line from the coordinates of two points on it

37

Unit 5: Trigonometry

5.1 Triangles

Use triangle inequality

Use Pythagoras’ theorem

Apply the sine, cosine and tangent ratios to the calculation of a side or of an angle of a

right-angled triangle

Recall and use the exact values of trigonometrical functions of special angles

Solve problems using the sine and cosine rules and the formula cabsin2

1 for the area

of a triangle

Know the range of values of for which cos is positive or negative

Triangle inequality

In any triangles ABC, the sum of the lengths of two sides is greater than the length of the

third side. This is known as the triangle inequality i.e.

AB < BC + AC

BC < AB + AC

AC < AB + BC

Simple trigonometrical ratios of an acute angle

c

a

hypothenus

oppositesin

c

b

hypothenus

adjacentcos

b

a

adjacent

oppositetan

The signs of the trigonometrical ratios

The trigonometric ratio of special angles

y

x

1st Quadrant

ALL positive

4th

Quadrant

cos positive

2nd

Quadrant

sin positive

3rd

Quadrant

tan positive

C

A B

a

b

c

A

B C

c

a

b

38

0 30 45 60 90

sin 0

2

0

2

1

2

1

2

2

2

3 1

2

4

cos 1

2

4

2

3

2

2

2

1

2

1 0

2

0

tan 0

3

1 1 3

Pythagoras’ theorem

In a right-angled triangle, the square of

the hypotenuse is equal to the sum of the

squares of the other two sides i.e.

222 BCABAC

222 abc

Sine rule

In any triangle ABC, RC

c

B

b

A

a2

sinsinsin ,

where R is the circumradius of the triangle.

Cosine rule

In any triangle ABC,

Abccba cos2222

Baccab cos2222

Cabbac cos2222

Area of a triangle

Area = heightbase2

1

Area = Cabsin2

1 = Abcsin

2

1 = Bacsin

2

1

Area = ))()(( csbsass , 2

cbas

(Heron’s formula)

C

A B

a

b

c

A

B C

c

a

b

39

Triangle Inequality

1. J97 Q23

The lengths of the sides of a quadrilateral are given by 1996 cm, 1997 cm, 1998 cm and z

cm. If z is an integer, what is the largest possible value of z?

2. J98 Q11

Find the total number of triangles such that the lengths (in cm) of all three sides of each

triangle are positive integers and the length of the longest side of each triangle is 15 cm.

3. J00 Q29

Determine the number of acute-angled triangles (i.e. all angles are less than 90) having

consecutive integer sides (say n – 1, n, n + 1) and perimeter not more than 2000.

Miscellaneous

1. J95 Q10

In the diagram, M is the midpoint of the semi-circular arc drawn on one side of a 6 cm by 7

cm rectangle. What is the perimeter of isosceles triangle MBC?

A

C

D

B

M

7 cm

6 cm

2. J95 Q18

Four congruent circles, each of which is tangent externally to two of the other three circles,

are circumcised by a square of area 144 cm2. If a small circle is then placed in the center so

that it is tangent to each of the circles, what is the diameter of this small circle?

40

3. J95 Q19

ABCD is an isosceles trapezium with AB parallel to DC, AC = DC and AD = BC. If the

height of the trapezium is equal to AB, find the ratio of AB: DC.

D

B

C

A

4. J95 Q27

In a right angled triangle, the lengths of the adjacent sides are 550 and 1320. What is the

length of the hypotenuse (correct to the nearest whole number)?

5. J95 Q29

An isosceles right-angled triangle is removed from each corner of a square piece of paper so

that a rectangle remains. What is the length of a diagonal of the rectangle if the sum of the

areas of the cut-off pieces is 200 cm2?

6. J96 Q3

A rectangle whose length is twice that of its breadth has a diagonal equal to that of a given

square. What is the ratio of the area of the rectangle to the area of the square?

41

7. J96 Q5

In the following figure, 6 right-angled triangles are assembled together. Given that PQ =

a and QR = 8a, find the expression (b – a)(b + a), in terms of a.

Q P

b

aa

a

a

R

8. J96 Q12

In the diagram, CD = 10 cm, CE = 6 cm and DE = 8 cm. Find the area of the rectangle ABCD.

CD

A BE

9. J96 Q23

In the diagram, AB = BD = 5 cm, ABD = 90 and DC = 2AD. Calculate the length of BC.

B

A CD

10. J97 Q5

A girls’ camp is located 300 m from a straight road. A boys’ camp is located on this road and

its distance from the girls’ camp is 500 m. It is desired to build a canteen on the road which

shall be exactly the same distance from each camp. What is the distance of the canteen from

each of the camps?

42

11. J97 Q20

Triangle ABC has sides AB = AC = 13 cm and BC = 10 cm. Another triangle, PQR, has the

same area as ABC with PQ = PR = 13 cm. Given that the two triangles are not congruent,

calculate the length of QR.

12. J98 Q4

In the figure below, the ratio of the area of the quarter circle to that of the inscribed rectangle

is 21:50 . If the radius of the quarter circle is 5 cm, find the perimeter of the rectangle in

cm.

43

Unit 6 Combinatorics

6.1 Counting

Use the strategy of systematic listing/counting

Use the addition principle

Use the multiplication principle

Recognize and distinguish between a permutation case and a combination case

Know and use the notation n! and the expressions for permutations and combinations

of n items taken r at a time

Answer problems on arrangement and selection (can include cases with repetition of

objects, or with objects arranged in a circle or involving both permutations and

combinations)

1. J95 Q11

Each time the two hands of a certain standard 12-hour clock form a 180 angle, a bell chimes

once. From noon today till noon tomorrow, how many chimes will be heard?

(A) 20 (B) 21 (C) 22 (D) 23 (E) 24

2. J97 Q29

How many numbers greater than ten thousand can be formed with the digits 0, 1, 2, 2, 3

without repetition? (Note that the digit 2 appears exactly twice in each number formed.)

3. J98 Q19

Seven identical dominoes of size 1 cm 2 cm and with identical faces on both sides are

arranged to cover a rectangle of size 2 cm 7 cm. One possible arrangement is shown in the

diagram below. Find the total number of ways in which the rectangle can be covered by the

seven dominoes.

4. J99 Q13

Two different numbers are to be chosen from the set {11, 12, 13, …, 33} so that the sum of

these two numbers is an even number. Find the number of ways to choose the two numbers.

44

5. J99 Q19

In a quiz containing 10 questions, 4 points are awarded for each correct answer, 1 point is

deducted for each incorrect answer and no point is given for each blank answer. The number

of possible scores is

(A) 10 (B) 40 (C) 44 (D) 45 (E) 50

6. J99 Q 20

The number of positive integers from 1 to 500 that can be expressed in the form ba with a

and b being integers greater than 1 is

(A) 25 (B) 27 (C) 29 (D) 33 (E) 35

7. J99 Q23

How many ways are there to form a three-digit even integer using the digits 0, 1, 2, 3, 4, 5

without repetition?

8. J00 Q12

How many numbers greater than 2000 can be formed by using some or all of the digits 1, 2, 3,

4, 5 without repetition?

9. J00 Q25

How many of the integers between 20000 and 29999 have exactly one pair of identical digits?

(Note: The two identical digits need not be next to each other. For example, 20130 is one of

the numbers we are looking for as it contains exactly one identical pair of digits, namely, 0;

whereas 20230 and 20030 are not.)

6.2 Graph Theory

Use graphs to model and solve problems

1. J95 Q28

Ten players took part in a round-robin tournament (i.e. every player must play against every

other player exactly once). There were no draws in this tournament. Suppose that the first

player won 1x games, the second player won 2x games, the third player 3x games and so on.

Find the value of

10987654321 xxxxxxxxxx .

45

2. J98 Q7

In a league competition which consists of 11 team, each team plays against every other twice.

Each match between two teams always results in a winner, and the winning team in each

match will be given an amount of $200 as prize-money. What is the total amount of prize-

money, in dollars, given out in the whole competition?

6.3 Pigeonhole Principle

Know and use the pigeonhole principle

46

Unit 7: Elementary Number Theory

7.1 Properties of Numbers

Know the (self-evident) properties of rational, irrational and real numbers

e.g. naa n for 1 ,1 If (J04 Q9)

e.g. even is 0 and 0 naa n (J00 Q19)

e.g. the cube of a fraction cannot be an integer unless the fraction is an integer

(J98 Q30)

1. J95 Q5

Consider the following statements:

(i) is a non-recurring decimal.

(ii) is an irrational number.

(iii) 722 .

(iv) is approximately 3.142.

(v) is a real number.

(A) Only (iii) is true. (B) Only (ii) and (iii) are true.

(C) Only (i), (ii), (iv) and (v) are true.

(D) Only (iii) and (iv) are true.

(E) Only (iii) and (v) are true.

7.2 Functions (Greatest Integer Function)

Know and use the greatest integer function

Know the properties of the greatest integer function

Use the result aaa }{][

1. J96 Q22

The symbol [x] is defined as the greatest integer less than or equal to the number x. If a [a] =

68 and b [b] = 109, what is the value of [a] [b] – [a + b]?

2. J00 Q30

Find the total number of integers n between 1 and 10000 (both inclusive) such that n is

divisible by ][ n . Here ][ n denotes the largest integer less than or equal to n .

47

7.3 Prime factorization

Determine the HCF and LCM of two or more numbers

Use prime factorization to determine the factors of a number

Note: Students should develop sensitivity towards numbers e.g. J02 requires

recognizing the common factors of seemingly unrelated numbers – 396 = 4(99), 297 =

3(99), 198 = 2(99)

Use prime factorization to determine the number of factors of a number (need

knowledge of combinatorics)

1. J96 Q19

Let a, b, c, d be integers, and 29))(( 2222 dcba . Find the value of 2222 dcba .

2. J96 Q30

The symbol n! is defined as n 321 . For example, 5! = 12054321 . Given

that 23191713117532! 22361323 n . What is the value of n?

3. J97 Q10

If x and x

221 are both integers, what is the total number of possible values of x?

4. J98 Q6

A card is chosen at random from a pack of 8 cards which are numbered 2, 3, 5, 7, 11, 13, 17,

19 respectively. The number of the card is recorded, and then the card is placed back with the

other cards. The cards are then shuffled, and the above process is repeated until a total of four

cards are chosen. Suppose the product of the four numbers thus obtained is P. How many of

the numbers 136, 198, 455, 1925, 3553 cannot be equal to P?

5. J98 Q9

Find the total number of positive integers x such that 324000 is divisible by x and x is

divisible by 20.

6. J98 Q21

372 identical cubes are placed together to form a rectangular solid. Find the total number of

different rectangular solids which can be formed in this way.

7. J99 Q15

The number of positive integers that are factors of

1)1636363(62 23

is

(A) 4 (B) 16 (C) 25 (D) 32 (E) 45

48

8. J00 Q8

How many (positive integer) factors does the number 1710 have?

(A) 289 (B) 290 (C) 323 (D) 324 (E) none of the above

9. J00 Q22

Let a, b, c, d be four distinct positive integers whose product abcd is equal to 2000. What is

the largest possible value of the sum a + b + c + d?

7.4 Modular Arithmetic

Know and use the Quotient – Remainder Theorem

Know and use the properties of modular arithmetic (e.g. J01 Q20 x sum of digits of

x (mod 9))

Know and use the periodic property of modular arithmetic to determine the last (few)

digit(s) of a number (Classic: J02 Determine the last digit of 20022002

)

Deduce the units digit of a number, a given the units digit of na for n = 1 or 2 or 3 etc

1. J95 Q4

What is the unit digit in ?)633)(163)(243( 8910

2. J95 Q12

Twenty soldiers, numbered 1 through 20, stood in a circle clockwise numerical order, all

facing the center. They began to count out loud in clockwise order: the first soldier called out

the number 1, the second called out 2; and each soldier then called out the numbers 1 more

than the number called to his right. What was the number of the soldier who called out the

number 1995?

3. J95 Q23

A natural number gives the same remainder (not zero) when divided by 3, 5, 7 or 11. Find the

smallest possible value of this natural number.

4. J97 Q19

A number x is divisible by the numbers 2, 3, 4, 5, 8 and 9, but leaves a remainder of 5 when

divided by 7. Find the smallest possible value of x.

5. J97 Q30

Find the smallest positive integer n such that 11001000 n and nnnn 1444133312221111 is divisible by 10.

49

6. J98 Q16

What is the unit digit of 199719991998 199919981997 ?

7. J00 Q11

What is the units digit of )17)(7)(3( 200120001999 ?

7.5 Divisibility

Know and use the divisibility tests for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13 and 25

Calculate the number of integers (within a given range) divisible by a certain number

(integer) (use of the greatest integer function is optional)

Use the divisibility property – the remainder of A when divided by n is the same as

the remainder of the sum of the digits of A when divided by n

Use the divisibility property – if na and ab , then nb

Use the divisibility property – if na and ab , then nab

Use the divisibility property – if cba and ba , then ca

Use the divisibility property – if nab , then na and nb

Use the divisibility property – if ma and mb leave the same remainder, then a – b is

divisible by m

Deduce the (unknown) numerator of a fraction (numerically equal to an integer) by

observing its denominator … etc

1. J96 Q24

Mr A, Mr B, Mr C and Mr D are car salesmen. In the period from 1985 to 1995, Mr A sold 8

times as many cars sold by Mr B, times as many sold by Mr C and 12 times as many sold by

Mr D. During this period, the total number of cars sold by the four salesmen was less than

600. What is the greatest possible number of cars which Mr A could have sold from 1985 to

1995?

2. J98 Q12

When the three numbers 1238, 1596, 2491 are divided by a positive integer m, the remainders

are all equal to a positive integer n. Find m + n.

3. J99 Q28

What is the smallest positive integer n such that the digits of n are either 0 or 1 and n is

divisible by 225?

50

7.6 Implicit Properties of Digits (of a number)

Recognize that each digit of number lies between 0 and 9 (inclusive)

7.7 Legendre’s Formula

Use the formula –

rp

n

p

n

p

n

2such that rr pnp 1 to determine the

exponent of the greatest power of a prime p dividing n!

(Classic example: Determine the number of zeros at the end of n!)

1. J00 Q18

How many (consecutive) zeros are there at the end of the number

10099321!100 ?

(For example, there are 2 (consecutive zeros) at the end of the number 30100.)

Practice

1. J95 Q23

A natural number (> 2) gives the same remainder (not zero) when divided by 3, 5, 7 or 11.

Find the smallest possible value of this natural number.

7.8 Diophantine equations

Solve diophantine equations

1. J97 Q22

A 2-digit number represented by BC is such that the product of BC and C is a 3-digit number

represented by ABC . Find all the possible2-digit numbers represented by BC.

2. J97 Q24

A solution of the equation 05))()(( cxbxax is x = 1, where a, b, and c are different

integers. Find the value of cba .

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3. J97 Q26

A rectangle has length p cm and breadth q cm, where p and q are integers. If p and q satisfy

the equation 213 qqpq , calculate the maximum possible area of the rectangle.

4. J97 Q27

Suppose x, y, and z are positive integers such that x > y > z > 663 and x, y and z satisfy the

following:

x + y + z = 1998

2x + 3y + 4z = 1998

5. J98 Q13

Suppose a, b are two numbers such that 06514822 baba . Find the value of 22 baba .

6. J98 Q18

The age of a man in the year 1957 was the same as the sum of the digits of the year in which

he was born. Find his age in the year 1998.

7. J98 Q20

Let x and y be two positive integers such that x – y = 75 and the least common multiple of x

and y is 360. Find the value of yx .

8. J98 Q28

Find the total number of positive four-digit integers x between 1000 and 9999 such that x is

increased by 2088 when the digits of x are reversed. [As an example, the integer 1234 is

changed to 4321 after reversing the digits.]

9. J98 Q29

Let a, b, c be positive integers such that ab + bc = 518 and ab – ac = 360. Find the largest

possible value of the product abc .

10. J98 Q30

Suppose a, b, c, d are four positive integers such that ,23 ba ,23 dc and 73 ac . Find

the value of ca .

11. J99 Q18

If a and b are positive integers such that 1522 ba and 2833 ba , then the number of

possible pairs of (a, b) is

(A) 0 (B) 1 (C) 2 (D) 3 (E) None of the above

52

12. J99 Q22

Suppose that p, q, (2p – 1)/q, (2q – 1)/p are positive integers and p, q > 1. What is the value

of qp ?

13. J00 Q3

A 4-digit number abcd consisting of 4 distinct digits satisfies

dcbaabcd 9 .

Then the second digit b is

(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

14. J00 Q21

Let x be a 3-digit number such that the sum of the digits equals 21. If the digits of x are

reversed, the number thus formed exceeds x by 495. What is x?

15. J00 Q23

One of the integers among 1, 2, 3, …, n is deleted. The average of the remaining n – 1

numbers is 17

602 . Which number was deleted?

16. J00 Q27

Let n be a positive integer such that n + 88 and 28n are both perfect squares. Find all the

possible values of n.