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8(abcd+1)>(a+1)(b+1)(c+1)(d+1) prove it

if a,b,c,d,all are greater than 1

Note by Aditya Kumar
4 years, 1 month ago

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What have you tried? What do you know?

As a hint : If \(x, y >1 \), then \( (x-1)(y-1) > 0 \) so \( xy + 1 > x + y \). Use this many times. Calvin Lin Staff · 4 years, 1 month ago

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Try to substitute \(x=a-1, y=b-1, z=c-1, w=d-1\). Shourya Pandey · 4 years ago

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@Shourya Pandey Don't you mean \(x=a+1,y=b+1\), etcetera? Tim Vermeulen · 4 years ago

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@Tim Vermeulen Try doing it, it becomes very easy Shourya Pandey · 4 years ago

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@Tim Vermeulen No Shourya Pandey · 4 years ago

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Just expand it , and try to put in square form , its very simple Shivang Jindal · 4 years, 1 month ago

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@Shivang Jindal I agree with just expand it. I'm not so sure what you mean by square form, though there is a way to write it as the sum of several positive (non-negative) terms. Calvin Lin Staff · 4 years ago

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@Shivang Jindal I think, its from Challenge and thrill . right? Shivang Jindal · 4 years, 1 month ago

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this question was from the beginning section of the book in inequalities so it does not require any prerequisities so i tried so solve this without am gm inequalitiy or any other but cannot solve it Aditya Kumar · 4 years, 1 month ago

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@Aditya Kumar As a hint : If \(x,y>1\), then \((x−1)(y−1)>0\) so \(xy+1>x+y\).

This looks very similar to the inequality, especially on the LHS. Use this many times. Calvin Lin Staff · 4 years ago

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thanks i have now solved it Aditya Kumar · 4 years ago

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