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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.
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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?
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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.