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
0proveeasier tobecanit,proveTo  yxyx

7. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi



8. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22

9. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22
2
2 22
qpqp 


10. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22
2
2 22
qpqp 

 
2
2
qp 

0

11. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22
2
2 22
qpqp 

 
2
2
qp 

0
pq
qp



2
22

12. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22
2
2 22
qpqp 

 
2
2
qp 

0
pq
qp



2
22
OR
2 2
Assume
2
p q
pq



13. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22
2
2 22
qpqp 

 
2
2
qp 

0
pq
qp



2
22
OR
2 2
Assume
2
p q
pq


2 2
2pq p q 
2 2
2
0 2
0 ( )
p pq q
p q
  
 

14. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22
2
2 22
qpqp 

 
2
2
qp 

0
pq
qp



2
22
OR
2 2
Assume
2
p q
pq


2 2
2pq p q 
2 2
2
0 2
0 ( )
p pq q
p q
  
 
2
But ( ) 0p q 

15. Inequality Techniques
0proveeasier tobecanit,proveTo  yxyx
  
2
Prove1995e.g.
22
qp
pqi


pq
qp


2
22
2
2 22
qpqp 

 
2
2
qp 

0
pq
qp



2
22
OR
2 2
Assume
2
p q
pq


2 2
2pq p q 
2 2
2
0 2
0 ( )
p pq q
p q
  
 
2
But ( ) 0p q 
2 2
2
p q
pq

 

16.    acbcabcbaii  222
Provea)1994

17.    acbcabcbaii  222
Provea)1994
  0
2
ba

18.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba

19.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222


20.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22



21.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22


bcacabcba 222222 222


22.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22


bcacabcba 222222 222

bcacabcba  222

23.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22


bcacabcba 222222 222

bcacabcba  222
3
1
prove,1Ifb)  bcacabcba

24.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22


bcacabcba 222222 222

bcacabcba  222
3
1
prove,1Ifb)  bcacabcba
   bcacabcbacba  2
2222

25.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22


bcacabcba 222222 222

bcacabcba  222
3
1
prove,1Ifb)  bcacabcba
   bcacabcbacba  2
2222
    bcacabbcacabcba  2
2

26.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22


bcacabcba 222222 222

bcacabcba  222
3
1
prove,1Ifb)  bcacabcba
   bcacabcbacba  2
2222
    bcacabbcacabcba  2
2
   2
3 cbabcacab 

27.    acbcabcbaii  222
Provea)1994
  0
2
ba
02 22
 baba
abba 222

bccb
acca
2
2
22
22


bcacabcba 222222 222

bcacabcba  222
3
1
prove,1Ifb)  bcacabcba
   bcacabcbacba  2
2222
    bcacabbcacabcba  2
2
   2
3 cbabcacab 
 
3
1
13


bcacab
bcacab

28.   3
3
1
Provec) abccba 

29.   3
3
1
Provec) abccba 
bcacabcba  222

30.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba

31.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba
   0222
 bcacabcbacba

32.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba
   0222
 bcacabcbacba
022322
2223222223


bcacabcccbca
cbabcabbcbbaabccabaacaba

33.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba
   0222
 bcacabcbacba
022322
2223222223


bcacabcccbca
cbabcabbcbbaabccabaacaba
03333
 abccba

34.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba
   0222
 bcacabcbacba
022322
2223222223


bcacabcccbca
cbabcabbcbbaabccabaacaba
03333
 abccba
  abccba  333
3
1

35.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba
   0222
 bcacabcbacba
022322
2223222223


bcacabcccbca
cbabcabbcbbaabccabaacaba
03333
 abccba
  abccba  333
3
1
3
1
3
1
3
1
,,let ccbbaa 

36.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba
   0222
 bcacabcbacba
022322
2223222223


bcacabcccbca
cbabcabbcbbaabccabaacaba
03333
 abccba
  abccba  333
3
1
3
1
3
1
3
1
,,let ccbbaa 
  3
1
3
1
3
1
3
1
cbacba 

37.   3
3
1
Provec) abccba 
bcacabcba  222
0222
 bcacabcba
   0222
 bcacabcbacba
022322
2223222223


bcacabcccbca
cbabcabbcbbaabccabaacaba
03333
 abccba
  abccba  333
3
1
3
1
3
1
3
1
,,let ccbbaa 
  3
1
3
1
3
1
3
1
cbacba 
  3
3
1
abccba 

38. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic




39. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic



    1prove,8111Supposed)  xyzzyx

40. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic



    1prove,8111Supposed)  xyzzyx
   
81
8111


xyzyzxzzxyyx
zyx

41. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic



    1prove,8111Supposed)  xyzzyx
   
81
8111


xyzyzxzzxyyx
zyx
  3
3
1
xyzzyx  AM GM

42. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic



    1prove,8111Supposed)  xyzzyx
   
81
8111


xyzyzxzzxyyx
zyx
  3
3
1
xyzzyx 
33 xyzzyx 
AM GM

43. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic



    1prove,8111Supposed)  xyzzyx
   
81
8111


xyzyzxzzxyyx
zyx
  3
3
1
xyzzyx 
33 xyzzyx 
   33 xzyzxyxzyzxy 
AM GM

44. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic



    1prove,8111Supposed)  xyzzyx
   
81
8111


xyzyzxzzxyyx
zyx
  3
3
1
xyzzyx 
33 xyzzyx 
   33 xzyzxyxzyzxy 
3 222
3 zyxxzyzxy 
AM GM

45. n
n
n
aaa
n
aaa


21
21
MeanGeometricMeanArithmetic



    1prove,8111Supposed)  xyzzyx
   
81
8111


xyzyzxzzxyyx
zyx
  3
3
1
xyzzyx 
33 xyzzyx 
   33 xzyzxyxzyzxy 
3 222
3 zyxxzyzxy 
 2
33 xyzxzyzxy 
AM GM

46. 81  xyzyzxzxyzyx

47. 81  xyzyzxzxyzyx
  8331
2
33  xyzxyzxyz

48. 81  xyzyzxzxyzyx
  8331
2
33  xyzxyzxyz
    8331
3
3
2
33  xyzxyzxyz

49. 81  xyzyzxzxyzyx
  8331
2
33  xyzxyzxyz
    8331
3
3
2
33  xyzxyzxyz
  81
3
3  xyz

50. 81  xyzyzxzxyzyx
  8331
2
33  xyzxyzxyz
    8331
3
3
2
33  xyzxyzxyz
  81
3
3  xyz
21 3  xyz

51. 81  xyzyzxzxyzyx
  8331
2
33  xyzxyzxyz
    8331
3
3
2
33  xyzxyzxyz
  81
3
3  xyz
21 3  xyz
13 xyz

52. 81  xyzyzxzxyzyx
  8331
2
33  xyzxyzxyz
    8331
3
3
2
33  xyzxyzxyz
  81
3
3  xyz
21 3  xyz
13 xyz
1xyz

53. OR
 1 2x x  AM GM

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

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

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

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

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

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

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

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

62.  
9 2 2 2 1 1 1
Proveiii
a b c a b b c a c a b c
     
    
 
1 1 1
2 2
4
a b ab
a b ab
 
    
 

1 1 4
a b a b
 

1 1 4
1 1 4
b c b c
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
Proveiii
a b c a b b c a c a b c
     
    
 
1 1 1
2 2
4
a b ab
a b ab
 
    
 

1 1 4
a b a b
 

1 1 4
1 1 4
b c b c
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
Proveiii
a b c a b b c a c a b c
     
    
 
1 1 1
2 2
4
a b ab
a b ab
 
    
 

1 1 4
a b a b
 

1 1 4
1 1 4
b c b c
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
    
  
  3 3
1 1 1 1
3 3
9
a b c abc
a b c abc
 
      
 


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

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

67.  
9 2 2 2 1 1 1
Proveiii
a b c a b b c a c a b c
     
    
 
1 1 1
2 2
4
a b ab
a b ab
 
    
 

1 1 4
a b a b
 

1 1 4
1 1 4
b c b c
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
    
  
  3 3
1 1 1 1
3 3
9
a b c abc
a b c abc
 
      
 

1 1 1 9
a b c a b c
  
 
 
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b 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
Proveiii
a b c a b b c a c a b c
     
    
 
1 1 1
2 2
4
a b ab
a b ab
 
    
 

1 1 4
a b a b
 

1 1 4
1 1 4
b c b c
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
    
  
  3 3
1 1 1 1
3 3
9
a b c abc
a b c abc
 
      
 

1 1 1 9
a b c a b c
  
 
 
1 1 1 9
2
2 2 2 9
a b b c a c a b c
a b b c a c a b c
  
    
  
    
9 2 2 2 1 1 1
a b c a b b c a c a b c
     
    
Inequalities Sheet
Exercise 10D
Note: Cambridge 8H (Book 1); 28