Simple inequalities

Number Theory Level 4

If a.b.c.d=1 Then (1+a).(1+b).(1+c).(1+d)>A Where a,b,c,d are Positive real nos. and A is a natural no. Find A's maximum value.


The answer is 15.

Kapil Chandak by Kapil Chandak , 21, India

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1 solution

Harsh Shrivastava
Nov 21, 2014

There should be a less than or equal to sign.

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It's an modification of an R.M.O question according to which we need yo prove (1+a).(1+b).(1+c).(1+d)>=16.So, ans. is 15 as we have to find only of > and not >= and A is natural no.

Kapil Chandak - 6 years, 6 months ago

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This can also be generalized, one can prove by AM-GM that ∏ k = 1 n ( 1 + a k ) ≥ 2 n \prod_{k=1}^{n} (1+a_k) \geq 2^n

Ryan Tamburrino - 6 years, 4 months ago

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