Number 4

Algebra Level 2

If a , b , c a,b,c and d d are 4 positive real numbers satisfying a b c d = 1 abcd = 1 , what is the minimum value of ( 1 + a ) ( 1 + b ) ( 1 + c ) ( 1 + d ) (1+a)(1+b)(1+c)(1+d) .


The answer is 16.

Soumya Shrivastva by Soumya Shrivastva , 20, India

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1 solution

Sam Bealing
May 16, 2016

Using repeated AM-GM :

( 1 + a ) ≥ 2 a (1+a) \geq 2 \sqrt{a}

( 1 + a ) ( 1 + b ) ( 1 + c ) ( 1 + d ) ≥ 16 a b c d = 16 (1+a)(1+b)(1+c)(1+d) \geq 16 \sqrt{abcd} =\boxed{16}

With equality when a = b = c = d = 1 a=b=c=d=1 .

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