Singapore Junior Math Olympiad 95 00 Questions

8/12/2019 Singapore Junior Math Olympiad 95 00 Questions

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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) Atypical 3 variable problem: Find the number of people required to complete a certainnumber of jobs in a certain number of days e.g. (J04 Q17)

1. J95 Q8What 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 Q3In the diagram, calculate the ratio of the area of the shaded region to the total area of the twoidentical smaller circles.

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

2. J97 Q25Three boys, Tom, John and Ken, agreed to share some marbles in the ratio of 9 : 8 : 7respectively. John then suggested that they should share the marbles in the ratio 8 : 7 : 6instead. Who would then get more marbles than before and who would get less than before ifthe ratio was changed?


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3. J99 Q6The 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 formulatime

distancespeed

Use the formulatimetotal

distancetotalspeedaverage

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, andthe second can fill the pool in four hours. There is also a drain that can empty the pool in sixhours. Both pipes were being used to fill the pool. After an hour, a careless maintenance manaccidentally opened the drain. How long more will it take for the pool to fill?

2. J96 Q4TownAand TownBare linked by a straight road. A factory is sited along the road such thatit is twice as far away from TownAas its distance from TownB. A truck left TownBat 9.00am and reached the factory 1 hour later. A car which travels three times faster than the truckneed 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 Q14If John walks from home to school at the speed of 4 km per hour, and walks back at the speedof 3.5 km per hour, find the average speed in km per hour for the whole trip.

(A) 3.75 (B) 3.6 (C)15

113 (D)

5

43 (E)

3

23

A B C


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

1. J95 Q16Two 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 Q17In a river with a steady current, it takes Bionic Woman 6 minutes to swim a certain distanceupstream, but it takes her only 3 minutes to swim back. How many minutes would it take adoll 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 Q30Two men are walking at different steady paces upstream along the bank of a river. A shipmoving downstream at constant speed takes 15 seconds to pass the first man. Five minuteslater it reaches the second man and takes 10 seconds to pass him. Starting then, how long willit 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 twoelements of the set and returns a single element. An example would be integer multiplication

"" where a, bare both integers and abreturns an integer.

1. J97 Q9

The operation is defined by: ab= a2b2.

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 ab= ab.Consider the following identities:

(i) ab= ab

(ii) (ab) c= a(bc)(iii) a(b+ c) = (ab)+(ac)


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

(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 logicaldeductions to evaluate the truth of statements

1. J00 Q20Four people,A,B, CandDare accused in a trial. It is known that

ifAis guilty, thenBis guilty;

ifBis guilty, then Cis guilty orAis not guilty;

ifDis guilty, thenAis guilty and Cis not guilty;

ifDis guilty, thenAis guilty.How many of the accused are guilty?

(A) 1 (B) 2 (C) 3 (D) 4(E) Insufficient information to determine


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Unit 3: Algebra

3.1 Algebraic representation and formulae

Use letters to express generalized numbers and express arithmetic processesalgebraically

Use the strategy: Students should note some questions need not be solvedalgebraically. 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). (Studentsshould 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 theresult 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 , 22

22

1)

1( x

xx

x (e.g. J04 Q23, J01 Q21) (Note:

to solve some questions, repeated use of this identity is necessary (e.g. J04 23, J01Q21))

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 theminimum or maximum value of expression)

Expansion and Factorisation

Some useful identities


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))((

))((

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) ofxmeans the numerical value ofx, 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

0if

0if2

aa

aaa (or aa 2 )

1. J95 Q6The 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)2

1 (E)4

1

2. J95 Q24

Ifxandysatisfy 722 yx , find the maximum value of 422 22 xyx .

3. J96 Q15If the value of 76x19yis 114, the value of 36x9yis

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


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4. J96 Q16

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

(A) 1 (B) 1 (C) 2a1 (D) 12a (E) None of the above

5. J96 Q18

If 51

x

x , find the value ofx

x 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 Q14Three boys agree to divide a bag of marbles in the following manner. The first boy takes onemore than half the marbles. The second takes a third of the number remaining. The third boyfinds that he is left with twice as many marbles as the second boy. The original number ofmarbles

(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 Q15Coffee 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 ofAis increased by 10% while that of Bis decreased by 15%, the cost ofthe mixture per kg remains unchanged. Findx:y.

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

9. J98 Q26Let mand nbe two integers such that 698 mnnm . Find the largest possible value of m.

10. J98 Q27Find the largest value ofxwhich satisfies the equation

222222 )73()2()2)(54()54( xxxxxxxx .


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11. J99 Q2If we increase the length and the width of a rectangle by 10 cm each, the area of the rectanglewill 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 Q9For any real numbers a, b and c, find the smallest possible value the following expression cantake:

23730185273 222 cabcba .

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

15. J00 Q13In the following diagram, ABCD is a rectangle and ADEF, CDHG, BCLM and ABNO aresquare. Suppose the perimeter ofABCDis 16 cm and the total area of the four squares is 68cm2. Find the area ofABCDin cm2.

N

AO H

CB

M

F

G

L

E

D

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


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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 canusually 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. frompolynomial of high degree to trigonometric function S01 Q25, from surds to quadraticJ98 Q15, from exponential to quadratic J96 Q11)

1. J95 Q13When some people sat down to lunch, they found there was one person too many for each tosit at a separate table, so they sat two to a table and one table was left free. How many tableswere there?

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

2. J95 Q21John 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 hisexcess luggage and the second paid $24 for his excess luggage. Had all the luggage belongedto one person, the excess luggage charge would have been $72. At most how many kilogramsof luggage is each person permitted to bring on the plane free of additional charge?

4. J96 Q11

Ifxandysatisfy 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


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5. J98 Q1A bag contains 28 marbles which are coloured either red, white, blue or green. There are 4more red marbles than white ones, 3 more white marbles than blue ones and 2 more bluemarbles than green ones. Find the number of white marbles.

6. J98 Q15It is given that a, b are two positive numbers satisfying

)5(3)( babbaa .

Find the value ofbaba

baba

223.

7. J99 Q4

Let x denote the absolute value of a numberxdefined 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 Q24When 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 Q16A student has taken nexaminations and 1 more examination is upcoming. If he scores 100 inthe upcoming examination, his overall average (of the n+ 1 examinations) will be 90; and ifhe 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


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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) ofxmeans the numerical value ofx, not considering its sign,

and is denoted by x .

0if

0if

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 Q29Let aand bthe two real roots of the quadratic equation

043)1( 22 kkxkx

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

2. J00 Q24Suppose the equation

01)4(2 axax

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


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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 bax

1. J95 Q1

The simplest expression for20

40

4

2is

(A) 1 (B) 4 (C)

20

2

1

(D) 202 (E) 182

2. J95 Q7What is the value ofxwhich satisfies

19952 19952 19952 19952 19952 19952 19952 19952 = x2 ?

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

3. J96 Q9Suppose 19961996 19961996 19961996 = x1996 , what is the value ofx?

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 ofx.

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


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6. J97 Q11

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

ab.

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

7. J00 Q2

The fifth root of5

55 is

(A) 55 (B) )15( 5

5 (C)

54

5 (D)4

55 (E)

55

5

8. J00 Q19How many integersolutions does the following equation have?

11 20002 xxx .

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

3.8 Standard Form

Use the standard form nA 10 where nis 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 thestandard 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 is38047.0 equal to?

(A) 0.521077119823 (B) 52.1077119823 (C) 521077.119823(D) 0.00521077119823 (E) 0.052107119823

2. J96 Q7The 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


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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 1stand 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 techniquecancellation 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 Classiccase: (J02 209 + 99 + 999 + = 10 1 + 1001 + 10001 +)

extrapolating the result of simple arithmetical calculation to cases involvinglarge numbers (e.g. J04 Q8What is the sum of all the digits in the number

2004102004 ?)

1. J95 Q22

Evaluate 123458123456123457

2469122

.

2. J96 Q14

Evaluate180018

)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 Q4Which 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


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

6. J97 Q12The 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 Q3Find the value of

88

44222

248252

)248252()248252(1000

.

8. J98 Q8

Find the value of 222222 199819974321 .

9. J98 Q10Find the value of

199819971997199719971998 .

10. J99 Q21

What is the product of

20002000

11)

1002

11(

1001

111001

222

?

11. J00 Q1Letxbe the sum of the following 2000 numbers:

44444,44,,4 .

Then the last four digits (thousands, hundreds, tens, units) ofxare

(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


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

zyyz or zxy 2 or2

zxy

(arithmetic mean)

Geometric Progression (GP)

The nth term of a GP (with common ratio r) is given by1

1

nn 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 Q171

1 11 2 1

1 3 3 11 4 6 4 1

etc.

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

second row is two 1s, each row begins and ends with 1, and the kth integer in any row whenit is not 1, is the sum of the kth and (k1)th numbers in the immediately proceeding row. Findthe ratio of the number of integers in the first nrows which are not 1s and the number of 1s.

(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


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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 1317 16 15 14

In which columns will the number 1000 fall?

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

4. J97 Q21In 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 potatoesone by one into the basket, if he starts from the basket and run at an average speed of 6 m asecond?

5. J98 Q24

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

the value100a .

6. J99 Q1The 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 Q11The 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

nis !/3

100n

n

largest?


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9. J00 Q5Consider the following array of numbers:

A B C D E

2 5 8

23 20 17 14 1126 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 Q6Find the sum of all positive integers which are less than or equal to 200 and not divisible by 3or 5.

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

11. J00 Q10

For which positive integer kdoes the expressionk

k

001.1

2

have the largest value?

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


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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 for1,1If (J04 Q9)

e.g. 10

ba

afor 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 nis 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. trialand error) (e.g. manipulating inequalities J98 Q22), (involving greatest integerfunctions 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 sameexponent, J01 rewriting numbers as fractions with the same numerator but different

denominator using the identityyx

yxyx

, J96 by evaluating their

differences/ considering specific cases Determine the intersection of solution sets of at least 2 inequalities

1. J95 Q3Ifxis 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


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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, cand dare 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 Q27Find all possible real valuesysuch that 16847 yx and 12135 yx .

5. J97 Q28

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

6. J98 Q5

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

7. J98 Q22

What is the smallest positive integer nsuch that 02.03414 nn ?

8. J99 Q3

Suppose 26a 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 .


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10. J99 Q8 (properties of fraction)The integer part of the fraction

1999

1

1985

1

1984

1

1

is

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

11. J99 Q10Let aand bbe 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 forxinto identities by observation (usually to findsolutions of (linear combinations of) coefficients)

Identities

)()( xQxP )()( xQxP for all values ofx

To find unknowns in an identity,(a) substitute values ofx, or(b) equate coefficients of like powers ofx.

1. J00 Q17

If 2002002

210

1002 2)733( xaxaxaaxx , find the value of

20019886420 2222222 aaaaaaa .

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


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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 11

+ 222 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 inascending order. The sum of the powers of a and b in each term of the expansion is alwaysequal to n.

1rT =rrn

r

nbaC

If a = 1,n

b)1( = 1+ bCn 1 +2

2bCn + 3

3bCn + + nb

1rT = rrn 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 ABChas 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 fromPtoAB,ACandBC?

2. J95 Q26In the diagram, congruent radiiPS and QRintersect tangent SR. If the two disjoint shadedregions have equal areas and ifPS= 10 cm, what is the area of quadrilateralPQRS?


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4.2 Radian measure

Solve problems including arc length and sector of a circle, including knowledge anduse 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 Q8In the diagram,ABCDis a rectangle withAD= 2AB.MandNare midpoints ofADandBC

respectively. TriangleABEis an equilateral triangle. Calculate MEN.

A

CB

DM

N

E

2. J96 Q29In the following figure,AB=AC=BD. Findyin terms ofx.

B

A

CD

x y


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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 Q9A four-sided closed figure has opposite sides equal in length. Which of the followingstatements 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 withABparallel toDC. The pointEon CDis such that DAE=

BAEand CBE= ABE. Given thatAD= 13 cm andBC= 12 cm, calculate the length ofCD.

4.5 Polygons

Calculate interior and exterior angles of polygons

1. J99 Q5The sum of the angles

A+ B+ C+ D+ E+ F+ Gin the diagram is

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

G

F

ED

C

B

A


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

A ri ght cir cular conecan be generated by the rotation of the right-angled triangle VOCaboutVO, which represents the height of the cone. Every point on the circumference of the base isthe same distance lfrom the vertex V. The length lis called the slant height.

Answer the following questions before you proceed to deduce the formula for the curvedsurface 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 C1C2in the sector?

____________________

l

ll

h

r

V

C

O

V

V

C1

C1

C2

C2

Cut along VC


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Now, fill in the blanks:

Area of sector Arc length

Area of circle Circumference 360

down an expression for

1. J99 Q27The following diagram shows a solid cube of volume 1 cm3. Let M be the midpoint of the

edgeBC. What is the shortest distance in cm for an ant crawl from the vertex AtoM?

E

H

F

A

G

C

M

B

D

Given that

360nceCircumfere

lengthArc , write down the ratio of

360

in terms of rand l.

circleofAreanceCircumfere

lengthArcsectorofArea

Curved surface area of a cone =____________________

Total surface area of a closed cone = _________ +_________


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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 achord is perpendicular to the chord2. equal chords are equidistant from the centre of a circle

1. Tangent perpendicular to radius2. 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 quadrilateral2. exterior angle of cyclic quadrilateral


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Alternate segment theorem

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

1. J97 Q13In the diagram, AM =MB=MC= 5 andBC= 6. Find the area of triangleABC.

A

C

BM

2. J98 Q14In the figure below,A,B, C,Dare four points on a circle, and the line segmentsBAand CD

are extended to meet at the pointE. Suppose E= 42, and the arcsAB,BCand CDall have

equal lengths. Find the measure of BAC+ ACDin degrees.

B

C

E

A

D


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3. J00 Q26In the diagram below,A,B, C,Dlie on the line segment OE, andACand CEare diameters ofthe circles centred atBandDrespectively. The line OFis tangent to the circle centred at Dwith the point of contactF. If OA= 10, AC= 26 and CE= 20, find the length of the chordGH.

O A B DC

G

H F

E

4.8 Loci

Use the following loci and method of intersecting loci

sets of points in two dimensions which are equidistant from two givenintersecting straight lines

1. J97 Q16Line l2 intersects l1and line l3is parallel tol1. 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 theoremratio of area of triangles with common height = ratio of bases


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1

B

3

N

C

A

M

3 5

Find the ratio of the area of triangleMNC: area of triangleABC.

3. J96 Q13A quadrilateral has sides of length 4 cm, 6 cm, 8 cm and 9 cm respectively. Another similarquadrilateral has a side of length 12 cm. What is the largest possible perimeter of this similarquadrilateral?

4. J97 Q6In the diagram, the radii of the sectors OPQand ORSare 5 cm and 2 cm respectively. Findthe ratio of the area of the shaded region to the area of the sector OPQ.

5. J98 Q17

In the figure below BAC= 90andDEFGis a square. If the length ofBCis6

185and the

area ofABCis 1369, find the area of the squareDEFG.

A

B C

GD

FE

6. J98 Q23In the figure below,APis the bisector ofBAC, andBPis perpendicular toAP. Also,Kis themidpoint ofBC. Suppose thatAB= 8 cm andAC= 26 cm. Find the length ofPKin cm.

O

PQ

S R


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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

BPandACis parallel toRP. If the area of triangleBSPis 4 square units, find the

area of triangleABCin square units.

A

B C

R Q

S

P

8 J99 Q12In the diagram below, ABCD is a square and

n

m

HA

DH

GD

CG

FC

BF

EB

AE .

9. J99 Q17In triangleABC,D,EandFare points on the sidesBC,ACanABrespectively such thatBC=4CD,AC= 5AEandAB= 6BF. Suppose the area ofABCis 120 cm2, what is the area ofDEFin cm2?

AH


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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

1for 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 thethird side. This is known as the triangle inequality i.e.

AB < BC + ACBC < AB + ACAC < 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

1stQuadrant

ALL positive

4t Quadrant

cos positive

2n Quadrant

sin positive

3r Quadrant

tan positive

C

A B

a

b

c

A

B C

c

a

b


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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 nitems taken rat 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 andcombinations)

1. J95 Q11

Each time the two hands of a certain standard 12-hour clock form a 180angle, a bell chimesonce. From noon today till noon tomorrow, how many chimes will be heard?

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

2. J97 Q29How many numbers greater than ten thousand can be formed with the digits 0, 1, 2, 2, 3without repetition? (Note that the digit 2 appears exactly twice in each number formed.)

3. J98 Q19Seven 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 thediagram below. Find the total number of ways in which the rectangle can be covered by theseven dominoes.

4. J99 Q13Two 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.


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5. J99 Q19In a quiz containing 10 questions, 4 points are awarded for each correct answer, 1 point isdeducted for each incorrect answer and no point is given for each blank answer. The numberof 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 aand bbeing integers greater than 1 is

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

7. J99 Q23How many ways are there to form a three-digit even integer using the digits 0, 1, 2, 3, 4, 5without repetition?

8. J00 Q12How 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 ofthe 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 Q28Ten players took part in a round-robin tournament (i.e. every player must play against everyother player exactly once). There were no draws in this tournament. Suppose that the first

player won1

x games, the second player won2

x games, the third player3

x games and so on.

Find the value of

10987654321 xxxxxxxxxx .


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2. J98 Q7In 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 eachmatch 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


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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 requiresrecognizing the common factors of seemingly unrelated numbers396 = 4(99), 297 =3(99), 198 = 2(99)

Use prime factorization to determine the number of factors of a number (needknowledge of combinatorics)

1. J96 Q19

Let a, b, c, dbe 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 andx

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

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 theother cards. The cards are then shuffled, and the above process is repeated until a total of fourcards are chosen. Suppose the product of the four numbers thus obtained is P. How many ofthe numbers 136, 198, 455, 1925, 3553 cannot be equal toP?

5. J98 Q9Find the total number of positive integers x such that 324000 is divisible by x and x isdivisible by 20.

6. J98 Q21372 identical cubes are placed together to form a rectangular solid. Find the total number ofdifferent rectangular solids which can be formed in this way.

7. J99 Q15The number of positive integers that are factors of

1)1636363(62 23

is

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


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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 Legendres Formula

Use the formula

r

p

n

p

n

p

n

2such that rr pnp 1 to determine the

exponent of the greatest power of a primepdividing n!

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

1. J00 Q18How 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 byBCis such that the product ofBCand Cis a 3-digit numberrepresented by ABC. Find all the possible2-digit numbers represented byBC.

2. J97 Q24

A solution of the equation 05))()(( cxbxax isx= 1, where a, b, and care different

integers. Find the value of cba .


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12. J99 Q22Suppose thatp, q, (2p1)/q, (2q1)/pare positive integers andp, q> 1. What is the value

of qp ?

13. J00 Q3A 4-digit number abcdconsisting of 4 distinct digits satisfies

dcbaabcd9 .Then the second digit bis

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

14. J00 Q21Let 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 exceedsxby 495. What isx?

15. J00 Q23One of the integers among 1, 2, 3, , n is deleted. The average of the remaining n 1

numbers is17

602 . Which number was deleted?

16. J00 Q27

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

possible values of n.

Documents

Singapore Junior Math Olympiad 95 00 Questions