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Let a,b,c,d be positive real numbers such that a*b*c*d=1. Show that (1 + a)*(1 + b)*(1 + c)*(1 + d) >= 16.

Let a,b,c,d be positive real numbers such that a*b*c*d=1. Show that (1 + a)*(1 + b)*(1 + c)*(1 + d) >= 16.

Grade:11

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Aritra,

 

(1+a)(1+b)(1+c)(1+d) = 1+∑a+∑ab+∑abc+abcd

Note ∑a has 4 terms....

∑ab has 6 terms

∑abc has 4 terms.

 

So 

(1+a)(1+b)(1+c)(1+d) = 1+∑a+∑ab+∑abc+abcd = 2+∑a+∑ab+∑abc

Now use AM-GM inequality for the remaining summation terms.

 

We have ∑a/4 ≥ (abcd)1/4

So ∑a/4 ≥ 1

 

And ∑ab/6 ≥ 1,

∑abc/4 ≥ 1.

 

So (1+a)(1+b)(1+c)(1+d) ≥ 2+4+6+4 = 16

So (1+a)(1+b)(1+c)(1+d) ≥ 16.

 

And that solves the problem.

 

Hope that helps.

 

Best Regards,

Ashwin (IIT Madras).

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