Question

If a, b,c are positive real numbers such that a+b+c=1, prove that (1 + 1/a) (1 + 1/b) (1 + 1/c) > 64 ​

Answer 1

Answer:

[(1+a)(1+b)(1+c)]

7

>7

7

a

4

b

4

c

4

i.e.,(1+∑a+∑ab+abc)>7(abc)

4/7

Since, AM>GM

a+b+c+≥3(abc)

1/3

ab+bc+ac≥3(a

2

b

2

c

2

)

1/3

Hence,

1+∑a+∑ab+abc≥1+3(abc)

1/3

+3(abc)

2/3

+abc1+∑a+∑ab+abc≥1+(abc)

1/3

+(abc)

1/3

+(abc)

1/3

+(abc)

2/3

+(abc)

2/3

+(abc)

2/3

+abc

So,

LHS≥7[1(abc)

1/3

(abc)

1/3

(abc)

1/3

(abc)

2/3

(abc)

2/3

(abc)

2/3

(abc)]

1/7

≥7[(abc)

9/3

(abc)]

1/7

≥7(abc)

4/7