14 Chapter 2
Practice
98 Problem If 0
a ≤ b, shew that 1
8 ·
b − a
2
b ≤
a + b 2
− √
ab ≤
1 8
· b − a
2
a
99 Problem Prove that if a
, b, c are non-negative real numbers then a
2
+ 1b
2
+ 1c
2
+ 1 ≥ 8abc
100 Problem The sum of two positive numbers is 100. Find their maximum possible
product.
101 Problem Prove that if a
, b, c are positive numbers then a
b +
b c
+ c
a ≥ 3.
102 Problem Prove that of all rectangles with a given perimeter, the square has the
largest area.
103 Problem
Prove that if 0 ≤ x ≤ 1 then x − x
2
≤ 1
4 .
104 Problem Let 0
≤ a,b,c,d ≤ 1. Prove that at least one of the products a1 − b
, b1 − c, c1 − d
, d1 − a is
≤ 1
4 .
105 Problem Use the AM-GM Inequality for four non-negative real numbers to prove
a version of the AM-GM for eight non-negative real numbers.
2.3 Identities with Cubes
By direct computation we find that x + y
3
= x + yx
2
+ y
2
+ 2xy = x
3
+ y
3
+ 3xyx + y 2.14
106 Example The sum of two numbers is 2 and their product 5. Find the sum of their cubes.
Solution: If the numbers are x , y then x
3
+ y
3
= x + y
3
− 3xyx + y = 2
3
− 352 = −22 .
Two other useful identities are the sum and difference of cubes, x
3
± y
3
= x ± yx
2
∓ xy + y
2
2.15
107 Example Find all the prime numbers of the form n
3
− 1, n a positive integer. Solution: As n
3
− 1 = n − 1n
2
+ n + 1 and as n
2
+ n + 1 1, it must be the case that n − 1 = 1, i.e., n = 2. Therefore, the
only prime of this form is 2
3
− 1 = 7 .
108 Example Prove that
1 + x + x
2
+ ··· + x
80
= x
54
+ x
27
+ 1x
18
+ x
9
+ 1x
6
+ x
3
+ 1x
2
+ x + 1 .
Solution: Put S = 1 + x + x
2
+ ··· + x
80
. Then S − xS = 1 + x + x
2
+ ··· + x
80
− x + x
2
+ x
3
+ ··· + x
80
+ x
81
= 1 − x
81
, or S1 − x = 1 − x
81
. Hence 1 + x + x
2
+ ··· + x
80
= x
81
− 1 x − 1
. Therefore
x
81
− 1 x − 1
= x
81
− 1 x
27
− 1 ·
x
27
− 1 x
9
− 1 ·
x
9
− 1 x
3
− 1 ·
x
3
− 1 x − 1
. Thus
1 + x + x
2
+ ··· + x
80
= x
54
+ x
27
+ 1x
18
+ x
9
+ 1x
6
+ x
3
+ 1x
2
+ x + 1 .
109 Example Shew that
a
3
+ b
3
+ c
3
− 3abc = a + b + ca
2
+ b
2
+ c
2
− ab − bc − ca 2.16
Practice 15
Solution: We use the identity x
3
+ y
3
= x + y
3
− 3xyx + y twice. Then
a
3
+ b
3
+ c
3
− 3abc =
a + b
3
+ c
3
− 3aba + b − 3abc =
a + b + c
3
− 3a + bca + b + c − 3aba + b + c =
a + b + ca + b + c
2
− 3ac − 3bc − 3ab =
a + b + ca
2
+ b
2
+ c
2
− ab − bc − ca If a
, b, c are non-negative then a + b + c ≥ 0 and also a
2
+ b
2
+ c
2
− ab − bc − ca ≥ 0 by 2.13. This gives
a
3
+ b
3
+ c
3
3 ≥ abc.
Letting a
3
= x , b
3
= y , c
3
= z, for non-negative real numbers x , y, z, we obtain the AM-GM Inequality for three quantities.
Practice
110 Problem If a
3
− b
3
= 24 , a − b = 2, find a + b
2
.
111 Problem Shew that for integer n
≥ 2, the expression n
3
+ n + 2
3
4 is a composite integer.
112 Problem If tan x + cot x = a
, prove that tan
3
x + cot
3
x = a
3
− 3a.
113 Problem AIME 1986 What is the largest positive integer n for which
n + 10 |n
3
+ 100?
114 Problem
Find all the primes of the form n
3
+ 1.
115 Problem
Solve the system x
3
+ y
3
= 126 ,
x
2
− xy + y
2
= 21 .
116 Problem Evaluate the sum
1
3
√ 1 +
3
√ 2 +
3
√ 4
+ 1
3
√ 4 +
3
√ 6 +
3
√ 9
+ 1
3
√ 9 +
3
√ 12 +
3
√ 16
.
117 Problem Find a
6
+ a
−6
given that a
2
+ a
−2
= 4 .
118 Problem Prove that
a + b + c
3
− a
3
− b
3
− c
3
= 3a + bb + cc + a 2.17
119 Problem ITT 1994 Let a
, b, c, d be complex numbers satisfying a + b + c + d = a
3
+ b
3
+ c
3
+ d
3
= 0 .
Prove that a pair of the a , b, c, d must add up to 0.
2.4 Miscellaneous Algebraic Identities
