1. Types of Proof

2. Types of Proof
1. Deductive Proof
Start with known facts and deduce what you are trying to prove.

3. Types of Proof
1. Deductive Proof
Start with known facts and deduce what you are trying to prove.
2. Inductive Proof
Assume what you are trying to prove and induce a solution.

4. Types of Proof
1. Deductive Proof
Start with known facts and deduce what you are trying to prove.
2. Inductive Proof
Assume what you are trying to prove and induce a solution.
3. Proof by Contradiction
Assume the opposite of what you are trying to prove and create a
contradiction.
Contradiction means the original assumption is incorrect, therefore
the opposite must be true.

5. Inequality Techniques

6. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0

7. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2

8. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p2  q2
 pq
2

9. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p2  q2 p 2  2 pq  q 2
 pq 
2 2

10. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p2  q2 p 2  2 pq  q 2
 pq 
2 2
 p  q 2

2
0

11. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p2  q2 p 2  2 pq  q 2
 pq 
2 2
 p  q2

2
0
p2  q2
  pq
2

12. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p q
2 2
p  2 pq  q
2 2 p2  q2
 pq  OR Assume pq 
2 2 2
 p  q2

2
0
p2  q2
  pq
2

13. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p q
2 2
p  2 pq  q
2 2 p2  q2
 pq  OR Assume pq 
2 2 2
 p  q2
2 pq  p 2  q 2

2 0  p 2  2 pq  q 2
0
0  ( p  q) 2
p2  q2
  pq
2

14. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p q
2 2
p  2 pq  q
2 2 p2  q2
 pq  OR Assume pq 
2 2 2
 p  q2
2 pq  p 2  q 2

2 0  p 2  2 pq  q 2
0
0  ( p  q) 2
p2  q2
  pq
2 But ( p  q) 2  0

15. Inequality Techniques
To prove x  y, it can be easier to prove x  y  0
p2  q2
e.g. i 1995 Prove pq 
2
p q
2 2
p  2 pq  q
2 2 p2  q2
 pq  OR Assume pq 
2 2 2
 p  q2
2 pq  p 2  q 2

2 0  p 2  2 pq  q 2
0
0  ( p  q) 2
p2  q2
  pq
2 But ( p  q) 2  0
p2  q2
 pq 
2

16. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac

17. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2

18. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0

19. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab

20. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc

21. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2ac  2bc

22. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2ac  2bc
a 2  b 2  c 2  ab  ac  bc

23. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2ac  2bc
a 2  b 2  c 2  ab  ac  bc
1
b) If a  b  c  1, prove ab  ac  bc 
3

24. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2ac  2bc
a 2  b 2  c 2  ab  ac  bc
1
b) If a  b  c  1, prove ab  ac  bc 
3
a  b  c  a  b  c   2ab  ac  bc 
2 2 2 2

25. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2ac  2bc
a 2  b 2  c 2  ab  ac  bc
1
b) If a  b  c  1, prove ab  ac  bc 
3
a  b  c  a  b  c   2ab  ac  bc 
2 2 2 2
 a  b  c   2ab  ac  bc   ab  ac  bc
2

26. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2ac  2bc
a 2  b 2  c 2  ab  ac  bc
1
b) If a  b  c  1, prove ab  ac  bc 
3
a  b  c  a  b  c   2ab  ac  bc 
2 2 2 2
 a  b  c   2ab  ac  bc   ab  ac  bc
2
3ab  ac  bc   a  b  c 
2

27. ii 1994 a) Prove a 2  b 2  c 2  ab  bc  ac
a  b   0
2
a 2  2ab  b 2  0
 a 2  b 2  2ab
a 2  c 2  2ac
b 2  c 2  2bc
2a 2  2b 2  2c 2  2ab  2ac  2bc
a 2  b 2  c 2  ab  ac  bc
1
b) If a  b  c  1, prove ab  ac  bc 
3
a  b  c  a  b  c   2ab  ac  bc 
2 2 2 2
 a  b  c   2ab  ac  bc   ab  ac  bc
2
3ab  ac  bc   a  b  c 
2
3ab  ac  bc   1
1
ab  ac  bc 
3

28. 1
c) Prove a  b  c   3 abc
3

29. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc

30. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0

31. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0
a  b  c a 2  b 2  c 2  ab  ac  bc   0

32. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0
a  b  c a 2  b 2  c 2  ab  ac  bc   0
a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c
 a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0

33. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0
a  b  c a 2  b 2  c 2  ab  ac  bc   0
a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c
 a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0
a 3  b3  c3  3abc  0

34. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0
a  b  c a 2  b 2  c 2  ab  ac  bc   0
a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c
 a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0
a 3  b3  c3  3abc  0
a  b  c   abc
1 3 3 3
3

35. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0
a  b  c a 2  b 2  c 2  ab  ac  bc   0
a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c
 a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0
a 3  b3  c3  3abc  0
a  b  c   abc
1 3 3 3
3
1 1 1
let a  a , b  b , c  c
3 3 3

36. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0
a  b  c a 2  b 2  c 2  ab  ac  bc   0
a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c
 a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0
a 3  b3  c3  3abc  0
a  b  c   abc
1 3 3 3
3
1 1 1
let a  a , b  b , c  c
3 3 3
1 1 1
1
a  b  c   a 3b 3 c 3
3

37. 1
c) Prove a  b  c   3 abc
3
a 2  b 2  c 2  ab  ac  bc
a 2  b 2  c 2  ab  ac  bc  0
a  b  c a 2  b 2  c 2  ab  ac  bc   0
a 3  ab 2  ac 2  a 2b  a 2 c  abc  a 2b  b3  bc 2  ab 2  abc  b 2 c
 a 2 c  b 2 c  c 3  abc  ac 2  bc 2  0
a 3  b3  c3  3abc  0
a  b  c   abc
1 3 3 3
3
1 1 1
let a  a , b  b , c  c
3 3 3
1 1 1
1
a  b  c   a 3b 3 c 3
3
1
a  b  c   3 abc
3

38. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n

39. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n
d) Suppose 1  x 1  y 1  z   8, prove xyz  1

40. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n
d) Suppose 1  x 1  y 1  z   8, prove xyz  1
1  x 1  y 1  z   8
1  x  y  xy  z  xz  yz  xyz  8

41. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n
d) Suppose 1  x 1  y 1  z   8, prove xyz  1
1  x 1  y 1  z   8
1  x  y  xy  z  xz  yz  xyz  8
1
 x  y  z   3 xyz AM  GM
3

42. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n
d) Suppose 1  x 1  y 1  z   8, prove xyz  1
1  x 1  y 1  z   8
1  x  y  xy  z  xz  yz  xyz  8
1
 x  y  z   3 xyz AM  GM
3
x  y  z  33 xyz

43. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n
d) Suppose 1  x 1  y 1  z   8, prove xyz  1
1  x 1  y 1  z   8
1  x  y  xy  z  xz  yz  xyz  8
1
 x  y  z   3 xyz AM  GM
3
x  y  z  33 xyz
xy  yz  xz  33  xy  yz  xz 

44. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n
d) Suppose 1  x 1  y 1  z   8, prove xyz  1
1  x 1  y 1  z   8
1  x  y  xy  z  xz  yz  xyz  8
1
 x  y  z   3 xyz AM  GM
3
x  y  z  33 xyz
xy  yz  xz  33  xy  yz  xz 
xy  yz  xz  33 x 2 y 2 z 2

45. Arithmetic Mean  Geometric Mean
a1  a2    an n
 a1a2  an
n
d) Suppose 1  x 1  y 1  z   8, prove xyz  1
1  x 1  y 1  z   8
1  x  y  xy  z  xz  yz  xyz  8
1
 x  y  z   3 xyz AM  GM
3
x  y  z  33 xyz
xy  yz  xz  33  xy  yz  xz 
xy  yz  xz  33 x 2 y 2 z 2
xy  yz  xz  3 xyz 
2
3

46. 1  x  y  z  xy  xz  yz  xyz  8

47. 1  x  y  z  xy  xz  yz  xyz  8
1  3 xyz  3 xyz   xyz  8
2
3 3

48. 1  x  y  z  xy  xz  yz  xyz  8
1  3 xyz  3 xyz   xyz  8
2
3 3
1  3 xyz  3 xyz    xyz   8
2 3
3 3 3

49. 1  x  y  z  xy  xz  yz  xyz  8
1  3 xyz  3 xyz   xyz  8
2
3 3
1  3 xyz  3 xyz    xyz   8
2 3
3 3 3
1  3 xyz   8
3

50. 1  x  y  z  xy  xz  yz  xyz  8
1  3 xyz  3 xyz   xyz  8
2
3 3
1  3 xyz  3 xyz    xyz   8
2 3
3 3 3
1  3 xyz   8
3
1  3 xyz  2

51. 1  x  y  z  xy  xz  yz  xyz  8
1  3 xyz  3 xyz   xyz  8
2
3 3
1  3 xyz  3 xyz    xyz   8
2 3
3 3 3
1  3 xyz   8
3
1  3 xyz  2
3 xyz  1

52. 1  x  y  z  xy  xz  yz  xyz  8
1  3 xyz  3 xyz   xyz  8
2
3 3
1  3 xyz  3 xyz    xyz   8
2 3
3 3 3
1  3 xyz   8
3
1  3 xyz  2
3 xyz  1
xyz  1

53. OR
1  x   2 x AM  GM

54. OR
1  x   2 x AM  GM
1  y   2 y
1  z   2 z

55. OR
1  x   2 x AM  GM
1  y   2 y
1  z   2 z
(1  x)(1  y ) 1  z   2 x 2 y 2 z
 8 xyz

56. OR
1  x   2 x AM  GM
1  y   2 y
1  z   2 z
(1  x)(1  y ) 1  z   2 x 2 y 2 z
 8 xyz
8  8 xyz

57. OR
1  x   2 x AM  GM
1  y   2 y
1  z   2 z
(1  x)(1  y ) 1  z   2 x 2 y 2 z
 8 xyz
8  8 xyz
1  1 xyz
xyz  1
xyz  1

58. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c

59. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1
 a  b      2 ab  2
a b ab
4

60. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1
 a  b      2 ab  2
a b ab
4
1 1 4
 
a b ab

61. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1
 a  b      2 ab  2
a b ab
4
1 1 4
 
a b ab
1 1 4
 
b c bc
1 1 4
 
a c ac

62. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1
 a  b      2 ab  2
a b ab
4
1 1 4
 
a b ab
1 1 4
 
b c bc
1 1 4
 
a c ac
2 2 2 4 4 4
    
a b c ab bc ac

63. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1
 a  b      2 ab  2
a b ab
4
1 1 4
 
a b ab
1 1 4
 
b c bc
1 1 4
 
a c ac
2 2 2 4 4 4
    
a b c ab bc ac
1 1 1 2 2 2
    
a b c ab bc ac

64. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1 1 1 1 1
 a  b      2 ab  2  a  b  c       3 3 abc  3 3
a b ab a b c abc
4 9
1 1 4
 
a b ab
1 1 4
 
b c bc
1 1 4
 
a c ac
2 2 2 4 4 4
    
a b c ab bc ac
1 1 1 2 2 2
    
a b c ab bc ac

65. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1 1 1 1 1
 a  b      2 ab  2  a  b  c       3 3 abc  3 3
a b ab a b c abc
4 9
1 1 4 1 1 1 9
    
a b ab a b c abc
1 1 4
 
b c bc
1 1 4
 
a c ac
2 2 2 4 4 4
    
a b c ab bc ac
1 1 1 2 2 2
    
a b c ab bc ac

66. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1 1 1 1 1
 a  b      2 ab  2  a  b  c       3 3 abc  3 3
a b ab a b c abc
4 9
1 1 4 1 1 1 9
    
a b ab
1 1a b 1 c a  b  c
9
1 1
 
4   
b c bc a  b b  c a  c 2a  b  c
1 1 4 2 2 2 9
    
a c ac ab bc ac abc
2 2 2 4 4 4
    
a b c ab bc ac
1 1 1 2 2 2
    
a b c ab bc ac

67. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1 1 1 1 1
 a  b      2 ab  2  a  b  c       3 3 abc  3 3
a b ab a b c abc
4 9
1 1 4 1 1 1 9
    
a b ab
1 1a b 1 c a  b  c
9
1 1
 
4   
b c bc a  b b  c a  c 2a  b  c
1 1 4 2 2 2 9
    
a c ac ab bc ac abc
2 2 2 4 4 4
    
a b c ab bc ac
1 1 1 2 2 2
    
a b c ab bc ac
9 2 2 2 1 1 1
     
abc ab bc ac a b c

68. 9 2 2 2 1 1 1
 iii  Prove      
abc ab bc a c a b c
1 1 1 1 1 1 1
 a  b      2 ab  2  a  b  c       3 3 abc  3 3
a b ab a b c abc
4 9
1 1 4 1 1 1 9
    
a b ab
1 1a b 1 c a  b  c
9
1 1
 
4   
b c bc a  b b  c a  c 2a  b  c
1 1 4 2 2 2 9
    
a c ac ab bc ac abc
2 2 2 4 4 4 Inequalities Sheet
    
a b c ab bc ac
1 1 1 2 2 2 Exercise 10D
    
a b c ab bc ac Note: Cambridge 8H (Book 1); 28
9 2 2 2 1 1 1
     
abc ab bc ac a b c