Dear students - The Mathematical Problem Solving site · PDF fileDear students . This material has been used in training of mathematically talented students in lower secondary in preparation

Dear students

This material has been used in training of mathematically talented students in lower secondary in

preparation for the SMO Junior.

It can also be useful for teachers to select some of these questions for use in upper secondary Math

classes as a Higher Order Thinking activity to challenge the students.

It is my belief that if Education is to be meaningful, it must make our students smarter. And, doing

mathematical problem solving is that surest way in this direction.

In any problem solving activity, there is really no fixed way to do. What is most important is to find

an idea, test that idea, modify that idea to suit the problem, etc. It involves the typical guess and

check to find a line of attack or breakthrough.

The questions contain in this booklet can be solved in many ways, but the heading in each lesson

gives you a hint on the direction to proceed.

Be patient with yourself. You will usually need some incubation time to find a good idea, guess and

test your idea until you succeed.

If you work through this material well enough, you will surely find yourself becoming smarter, that

I’m sure.

Do keep a handy book for all your works, successes or failures. You will be well reminded that you

just need one good idea to succeed, before that, there maybe many failures.

So, problem solvers, good luck!

Mr Ang. 2012

Btw, the material is copyrighted, so not for commercial use. You can get in touch with me on

[email protected] .

mailto:[email protected]

Content

Page:

1. Completing Squares 1

2. Replacing Variables 4

3. Equating Coefficients 7

4. Forward-Backward 10

5. Reducing degree 13

6. Expanding 16

7. Cut and paste 19

8. Proof by contradiction 22

9. Area method 25

10. By construction 28

11. By factorising 31

12. Equation & Functions 37

13. Case Analysis 41

14. Practice #1 45

15. Practice #2 47

16. Practice #3 49

17. Practice #4 50

18. Practice #5 52

Thinking lesson#1

Prepared by KL Ang 2006 1

Completing Squares

Ex 1. If x, y,z are real numbers, and 4 1 2 9x y z x y z , find the

value of , ,x y z .

Ex 2. Given that a, b, c are real numbers, and 2a bc , 2b ca , 2c ab . Show

that a b c .

Thinking lesson#1


Ex 3. Given that x y m , y z n , find the value of. 2 2 2x y z xy yz zx .

Ex 4. Calculate: 3 5

2 7 3 5

.

Thinking lesson#1


Ex 5. Given that 2 2 22 1 3 4 4 2 0x a x a ab b has real roots, find the

value of a, b.

Ex 6. Solve: 2 4 16 2 20 0x x x .

Thinking lesson#2


Replacing Variables

Ex 7.

1999

1

3

1

2

1

2000

1

3

1

2

11

2000

1

3

1

2

1

1999

1

3

1

2

11

Ex 8. Given that 9876504321

9876012345M ,

9876504322

9876012346N , determine which is larger.

Thinking lesson#2


Ex 9. Given that 0a , 0b , and 3 2a b ab , find the value of 2a ab b

a ab b

.

Ex 10. Factorize: 2

4z x x y y z .

Thinking lesson#2


Ex 11. Given that 2 2 2x yz y zx z xy , show that x y z or 0x y z .

Ex 12.

Solve

13 3 0

12 8 0

x x yy

x yy

.

Thinking lesson#3


Equating Coefficients

Ex 13. Given that 3 23 5x x x a can be divided by 2 3x x , find a.

Ex 14. Factorize: 2 22 8 2 14 3x xy y x y .

Thinking lesson#3


Ex 15. If

11

3 2 5 3 2 5

A B

x x x x

, find the value of A and B..

Ex 16. Find the value of constant m, so that inequality 2 6 9x mx has a solution

5x .

Thinking lesson#3


Ex 17. A quadratic function passes through 1,0 with line of symmetry at 3x .

The maximum value is 2. Find the quadratic function.

Ex 18. Given that 0 x y and yx 2016 , find the number of ordered pair

,x y .

Thinking lesson#4


Forward-Backward

Ex 19. In figure 1-1, given that 1 2 , C D , show that A F .

Ex 20. In figure 1-2, given that //AD BC , AD BC , point E, F are two points on AC

and AE CF . Show that EDC FBA .

Thinking lesson#4


Ex 21. In figure 1-3, 1O and 2O are centre of two circles, intersect at A and B. CD

tangents to the two circles. CB extends to intersect AD at E. DB extends to intersect

AC at F. Show that DE DA DB DF and //EF CD .

Ex 22. Let 1x , 2x , 3x , …, 9x are all positive integers and 1 2 3 9x x x x . Given

that 1 2 3 9 220x x x x , when 1 2 3 4 5x x x x x is at its maximum, find the

value of 9 1x x .

Thinking lesson#4


Ex 23. In figure 1-4, given that in ABC , 90ACB , CD AB at D, 1CD . If

the length AD and BD are the roots to the equation 2 0x px q and

tan tan 2A B . Find the equation.

Ex 24. Mr Lee, Mr Chow and Mr Ong each taught two of these subjects: Biology,

Physics, English, PE, History and Mathematics.

Given that:

(1) Physics teacher and PE teacher are neighbours.

(2) Mr Lee is the youngest.

(3) Mr Ong and the Biology teacher and PE teacher often walk

home together.

(4) Biology teacher is older than the Mathematics teacher.

(5) On weekends, English teacher, Mathematics teacher and Mr

Lee like to play basketball.

Who taught which two subjects?

Thinking lesson#5


Reducing degree

Ex 25. Given that 2 1 0x x , find the value of 4 32 2 4 5x x x .

Ex 26. Find the minimum value of 2

2

3 6 5

11

2

x x

x x

.

Thinking lesson#5


Ex 27. Solve: 1 5 1 3

1 7 3 5

x x x x

x x x x

.

Ex 28. Solve:

2 2

2 2

2 2 2 0

2 4 2 3 3 4 0

x xy y x y

x xy y x y

.

Thinking lesson#5


Ex 29. 2 0x px q has 2 real roots, a and b. 1I a b , 2 2

2I a b , …,

n n

nI a b . When 3n , find the value of 1 2n n nI pI qI .

Ex 30. If 4 28 8 2 5 0x a x a is truth for all x, find the range of the real

number a.

Thinking lesson#6


Expanding

Ex 31. Simplify:

6 4 3 3 2

6 3 3 2

.

Ex 32. Calculate: 1 1 1 1

4 28 70 700 .

Thinking lesson#6


Ex 33. Find the value of 1 1 1 1

11 2 1 2 3 1 2 3 4 1 2 3 10

.

Ex 34. Calculate: 2 6 12 90 .

Thinking lesson#6


Ex 35. Factorize: 3 26 11 6x x x .

Ex 36. Given that n is a positive integer, the roots to the equation

2 22 1 0x n x n are n and n , find the value of:

3 3 4 4 5 5 20 20

1 1 1 1

1 1 1 1 1 1 1 1

.

Thinking lesson#7


Cut and paste

Ex 37. In figure 1-8, //AB CD . Show that 360B D BED

Ex 38. In figure 1-13, trapezium ABCD has //AD BC , E and F are mid-points on AD

and BC respectively. 90B C , show that 1

2EF BC AD .

Thinking lesson#7


Ex 39. In figure 1-20, given that right-angled ABC , 90C , 3AC , 5BC ,

form a square on side AB. Let the centre of the square be O, find the length of the

segment OC.

Ex 40. In figure 1-21, rectangle ABCD has length a, width b, find the shaded area.

Thinking lesson#7


Ex 41. In figure 1-22, ABC has 120B , D , E are points on AC and AB

respectively. Given that 7AC , 60EDC , AE BC , 3 3

sin14

A , find the

area of quadrilateral DEBC.

Ex 42. Given a linear function 1

22

y x intersects x-axis at A, and y-axis at B.

Equation 2y ax bx c passes through A and B with its axis of symmetry on the left

side of y-axis, find the range of a; and if the quadratic equation has a min/max point at

M, and intersects x-axis at another point C when 1

16a , find the area of

quadrilateral ABMC.

Thinking lesson#8


Proof by contradiction

Ex 43. Show that 2 is not a rational number.

Ex 44. Show that when p and q are both odd numbers, 2 2 2 0x px q , where

2 2 0p q has 2 irrational roots.

Thinking lesson#8


Ex 45. In figure 1-24, in ABC , 30A , 1

2BC AB , show that 90C .

Ex 46. If 1 2 1 2p p q q , show that 2

1 1y x p x q , and 2

2 2y x p x q at least

one has an intersection with x-axis.

Thinking lesson#8


Ex 41. Show that there are infinitely many prime numbers.

Ex 48. Solve: 3 3 3 x x .

Thinking lesson#9


Area method

Ex 49. By geometrical construction, construct a triangle with the same area as the

convex quadrilateral in figure 1-31.

Ex 50. In figure 1-32, a line is drawn from each vertices to the opposing side that

intersects a common point O. ABC is divided into 6 smaller , of which the area of

the 4 s are as shown. Find the area of ABC .

Thinking lesson#9


Ex 51. Find the areas enclosed by 3

34

y x , 3

2y x with x-axis and another one

with y-axis. Also find the shortest distance from the origin to the line3

34

y x .

Ex 52. In figure 1-34, in the regular pentagon ABCDE, AP, AQ and AR are

perpendicular lines from point A to CD, CB and DE (or its extended lines). O is the

centre of the pentagon and 1OP , find the length of AO AQ AR .

Thinking lesson#9


Ex 53. Given that in ABC , 1 2 as shown in figure 1-35. Show that

AB BD

AC DC .

Ex 54. In figure 1-37, N and M are the mid-points on the diagonal AC and BD

respectively. Point E, F, G, H are the mid-points on AB, BC, CD, DA. Through M

and N, draw parallel line AC and BD intersect at O, joins OE, OF, OG, OH. Show

that OEBF OFCG OGDH OHAES S S S .

Thinking lesson#10


By construction

Ex 55. Simplify:

(a)

24

2

a c b c a b

a b c

;

(b)

2 2 2a b c

a b a c b a b c c a c b

.

Ex 56. Find the exact value of sin18 .

Thinking lesson#10


Ex 57. In figure 1-40, given that in ABC , AD is the median, a line through B

intersects AD at F, and AC at E, so that AE EF . Show that BF AC .

Ex 58. Let a, b, c are real, and 4 4 0a b c , 2 0a b c , compare the value of 2b and ac.

Thinking lesson#10


Ex 59. As shown in figure 1-48, given that PC BC , AB BC , //PC AB , and

45CPA . If 20AB m, find the length PC.

Ex 60. In a square with side 1, given 9 points in the square, show that there exists a

triangle with area not larger than 1

8.

Thinking lesson#11


By factorising

Ex 61. Simplify:

(a) 2 2 2 2 2

1 1 1 1 11 1 1 1 1

2 3 4 9 10

,

(b) 2 4 8 16 323 1 3 1 3 1 3 1 3 1 3 1 .

Ex 62. Given that (1) x, y, z are positive real numbers; (2) 3 3 3 3x y z xyz ,

If (1) is truth, can the conclusion in (2) be also truth?

If (2) is truth, can the conclusion in (1) be also truth?

Thinking lesson#11


Ex 63.

(1) Given that 2 1 0x x , find the value of 3 22 2 3x x x ,

(2) Given 1

3 2x

, find the value of 5 4 3 210 10 2 1x x x x x .

Ex 64. Given that 1 1

3x y , find the value of

2 3 2

2

x xy y

x xy y

.

Thinking lesson#11


Ex 65. Given that 2 25 5 8 2 2 2 0x y xy y x , find the value of x and y.

Ex 66. Given 0abcd , a, b, c, d are real numbers, and

2 2 2 2 22 0a b d b a c d b c . Show that b c

da b .

Thinking lesson#11


EX 67. Show that 1 3 5 99 1

2 4 6 100 10 .

EX 68. Given that 0abcd , 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2

a b c dM

a b c b c d c d a d a b

, show that 1 2M .

Thinking lesson#11


Inductive reasoning

Ex 69. Given that ABC has an area of 16 cm2. The midpoint on AB, BC, CA are

1A ,

1B , 1C respectively. Let the area of 1 1 1A B C be

1S ; the mid-points on 1 1A B ,

1 1B C ,

1 1C A be 2A , 2B , 2C respectively. And the area of 2 2 2A B C be 2S . Find the area of

1999S .

Ex 70. Write 19 as a sum of natural numbers such that the product of these numbers

is the largest.

Thinking lesson#11


Ex 71. Given that M, N are the 2 points on the slant of a right-angled CAB , where

90C , that divide segment AB into 3 equal lengths. If cosCM ,

sinCN , find the length of the side AB .

Ex 72. If x is a real number, find the least value of 2 22 2 4 8y x x x x .

Thinking lesson#12


Equation & Functions

Ex 73.

(1) find the value of 2 2 2 ,

(2) Show that 0.9 1 .

Ex 74. In the figure 2-1, given that BC is the diameter of the semicircle centre at O,

CB extends to P, PA tangents on an arc CB at A such that AD BC . If

: 4 :1CD BD , 2PA , find the length PB and PC.

Thinking lesson#12


Ex 75. Given figure 2-2, in the rectangle ABCD, P,Q are on AD and BC respectively.

PQ BC , and PQ PD PA . If 5AQ cm, 2DQ cm, find the area of

rectangle ABCD.

Ex 76. If 2 2 1 2 1 0x a x a has a real root , such that 0 1 , find the

range of a..

Thinking lesson#12


Ex 77. In figure 2-4, rectangle ABCD are with 3AB , 4BC , P is moving on the

diagonal AC, but not on C. PE PB and intersect AD on E. Is there minimum or

maximum value for area PEAB?

Ex 78. Given that 0x y z , 2xyz , find the least value of x y z .

Thinking lesson#12


Ex 79. In figure 2-5, town C and town B is 30 km apart. C is 40 km away from A.

AC is a waterway. The shipping cost by water is half of that by road. If BD is the

road, where should D be from A so that the total cost of shipping from A to B is

minimum?

Thinking lesson#13


Case Analysis

Ex 80. Given that 1ab a b , where a and b are integers, find the number of

ordered pair ,a b .

Ex 81. Solve: 2 22 2 0x x k x x .

Thinking lesson#13


Ex 82. Find the value of a so that 2

1 1

2 1 2 101 2 a x aa x x a

has

only one solution.

Ex 83. If 1y k x and 1k

yx

appear only in one of the four quadrants together

where 0,1k , find the range of k.

Thinking lesson#13


Ex 84. Given that 90AOB , OC is a ray, OM and ON bisect BOC and AOC

respectively, find MON .

Ex 85. In the quadrilateral ABCD, //AD BC , AB DC , AC and BD intersect at O

with 120BOC , 7AD , 10BD . Find the length BC and the area of ABCD.

Thinking lesson#13


Ex 86. In figure 2-18, a cone is sectioned along the plane VAB where V is the vertex

of the cone. The height and the radius of the cone are h and r respectively.

a. When 1h , 3

4r , find the maximum value of VABS ;

b. When 1h , 3r , find the maximum value of VABS

.

Thinking Practice #1

Prepared by KL Ang, 2006 45

1. Given that x and y are real, and satisfies 2 25 5 8 2 2 2 0x y xy y x , find the

value of x and y.

2. Given that a, b, c, d are non-zero real, and 2 2 2 2 22 0a b d b a c d b c ;

show that b c

da b .

3. Given a, b, c are integers, 2 2 2 48 4 6 12a b c a b c ; find the value of

1 1 1abc

a b c

.

4. Given that 1 1

2 2x x a

, find the value of 2 1x

x

.

5. Factorise

a. 1 2 3 4 120x x x x ,

b. 2

2 2 1x y xy x y xy .

6. Solve

a. 2

2 22 14 28 15x x x x ,

b. 4 3 26 5 38 5 6 0x x x x .



7. Solve

a. 2

2

6

1x x

x x

,

b. 0a x a x a

axa x a x

.

8. Solve

a. 2 24 2 3 9x x x x x ,

b. 2 22 4 3 2x x x x .

9. Solve the system of equations

4

1 3 4

y x

x y

.

10. Given that x, y, z are non-negative real that are no more than 1, if

1 1 1k x y x z x y , find the range of k.



1. By way of comparing coefficients, determine the polynomial division of

23 14 5 5x x x .

2. Factorize : 2 22 5 2 7 5 3x xy y x y .

3. Find the value of m for which 2 6x mx can be factorized into two rational factors

4. Given the identity: 2

2

2

Mx N c

x x x a x b

,

2 2

Mx N

x x

is a simplified faction, and

a b , a b c . Find the value of N.

5. Given that a first degree function intersects x-axis at A(-6,0), it intersects a directly

proportional function at B, where B is in the 2nd

quadrant. Point B is on the 4x line.

∆AOB has an area of 15. Find the equations of the directly proportional and the first degree

functions.

6. Given that the solution to 4

13

x a is the same as that of

2 10

3

x , find the range

of a.

7. 2 22 2 4 3y x m x m m , where 0m , m Z . It intersects points A

and B on the x-axis, with point A on the left of the origin, point B on the right of the origin.

(1) Find the equation of y,

(2) If a liner function y kx b passes through point A and intersects the above

curve at point C such that 10ABCS . Find the equation of this linear

function.

8. In figure 1-5, given that ABC and a circum-

circle with centre O, chord AE bisects BAC , BF is

tangent to the circle at B and F is the intersection of the

line CE. Show that BF BC CE CF .



9. In figure 1-6, given that //AB DC , AD is the diameter of the circle with centre at O.

90C , line BC intersects with the circle at E and F. Let AB a , BC b , CD c .

Show that tan BAF and tan BAE are the roots of equation 2 0ax bx c .

10. Mr Tan and Mr Soon are two families each has two children, all below 9 years old.

The four children each has a different age. As a friend of them introduces these families to

you:

1. Ah Ming is 3 years younger than his brother.

2. Hai-Tao is the oldest among the children.

3. Ah Fong is half the age of one of the children in Mr Tan’s family.

4. Chi-Zhu is 5 years older than the younger child in Mr Soon’s family.

5. Five years ago, each family has only one child.

Find the children in each family and their respective ages.



1. Given that 3 1x , find the value of the expression: 10 9 8 6 5 4 3 22 2 2 4 4 2 2 1x x x x x x x x x .

2. Given that x is real , and 0x , as 1

2xx

, find the value of: 7 4

8 4

2

11 1

x x x

x x

.

3. Solve the equation: 2

2 24 2 2 3 0x x x x .

4. Solve the equation: 3 2 3 2

2 2

7 24 30 2 11 36 45

5 13 2 7 20

x x x x x x

x x x x

.

5. Solve the system of equations:

2 2

2 2

2 2 4 96,

2 43.

x xy y x

x xy y y

6. Factorize: 3 22 5 6.x x x

7. Calculate: 1 1 1 1

2 1 2 3 2 2 3 4 3 3 4 100 99 99 100

.

8. Find the sum of:

1 1 1 1

1 3 2 4 3 5 2n n

.

9. Find the value of α in a triangle, given that 2 25sin sin 3cos 3 .

10. For every natural number n, parabola 2 2 2 1 1y n n x n x and x-axis

intersect at two points nA , nB . n nA B denotes the distance between these two points, find

the value of 1 1 2 2 1999 1999A B A B A B .



1. In figure 1-26, given that in quadrilateral ABCD, 3AB ,

4BC , 13CD , 12AD , 90B , find the area quadrilateral

ABCD .

2. In figure 1-27, given that in quadrilateral ABCD ,

90B D , 60A , 4AB , 5AD . Find the

value of BC

CD and ABD

ABC

S

S

.

3. In figure 1-28, given that in sector AOB, OA OB R , 90AOB , with OA as a

diameter with centre at M, forms an semi-circle. //MP OB and intersects at P on the arc

AB. Q is the intersection of arc OA and MP. Find the area of the shaded region.

4. In figure 1-29, given that PA is tangent to circle with centre O at A. Point D is in

the circle and PD intersects the circle at C. If 6PA , 3PC DC , 2OD , find the

radius of the circle.



5. In figure 1-30, in the rectangle ABCD, 3AB , 4BC , point P is moving along

the diagonal line AC(does not overlap point C), PE PB , PE intersects AD at E.

a. Let PC x , S is the area of the quadrilateral PEAB, find the function

between S and the independent variable x, and the range of value x;

b. When PE AE , find the area of PEAB;

c. Is there any maximum or minimum value for the area PEAB? Why?

6. Let a, b, c be real numbers, and 2 22

A a b

, 2 23

B b c

, 2 26

C c a

.

Show that among A, B, C, at least one of them is greater than 0.

7. Given that in the ABC , AB c , BC a , CA b , and 2 2 2c a b . Show that

90C .

8. If 2 4 4 3 0x ax a , 2 21 0x a x a , 2 2 2 0x ax a , at least one of

them has real roots, find the range of the value a.

9. 50 students on a field. Every two of them are different distance apart. Each one of

them has a dart in their hand. Each is to throw the dart at the nearest student to him. Show

that a student can get at most hit by 5 darts.

10. Lattice points are points with integer co-ordinates. Show that 211

5y x x

cannot pass through any lattice point.



1. In figure 1-52, given that a concave quadrilateral ABCD, construct a triangle with

the side BC, including C as one of the internal angles, that has the same area with the

quadrilateral ABCD.

2. In figure 1-53, given that in quadrilateral ABCD has an area of 1. Extend the lines

AB, BC, CD, DA to point A’, B’, C’, D’ respectively. Such that ' 2BA AB , ' 2CB BC ,

' 2DC CD , ' 2AD DA . Find the area of quadrilateral A’B’C’D’.

3. In figure 1-54, given that in the diamond shape, 30ABC , show that 2AB AC BD .



4. By area method, show the two following theorems in geometry:

a. Ceva Theorem- let O is a point in ABC , AO, BO, CO extends to intersect

the triangle at D, E, F. Show that 1.BD CE AF

DC EA FB

b. Menelaus Theorem- a straight line intersects ABC on its sides or extended

line at F, E, D. Show that 1.BD CE AF

DC EA FB

5. In figure 1-55, two pieces of land is divided by a walk pathC F G . If the walk

path is to be straight and still passes through point C, of course, the two areas must remain

the same, how should this path to be drawn?

6. In figure 1-56, given that in the ABC , E, F are points on

AC, AB respectively, and 1

2EBC FCB A , show that

BF CE .

7. In figure 1-57, given that AD is the median in ABC , and BAD DAC . Show

that AC AB . (a median is a line joining a vertex to the midpoint of the opposite side.)



8. In figure 1-58, P is a point inside the equilateral triangle ABC. Given that 3PC ,

4PA , 5PB , find the length of the side of the ABC .

9. Let 0 1x , 0 1y , show that

2 2 2 22 2 2 21 1 1 1 2 2x y x y x y x y .

10. A square is divided into 9 9 81 congruent squares. In each smaller square, a

number from 1, 2, 3, …,19,20 is randomly written. Show that there are two pairs of

squares that are symmetrical about the centre of the larger square that the two pairs of

numbers have the same sum.

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Dear students - The Mathematical Problem Solving site · PDF fileDear students . This material has been used in training of mathematically talented students in lower secondary in preparation