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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
3 years, 11 months 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. Staff · 3 years, 11 months ago

Try to substitute $$x=a-1, y=b-1, z=c-1, w=d-1$$. · 3 years, 11 months ago

Don't you mean $$x=a+1,y=b+1$$, etcetera? · 3 years, 11 months ago

Try doing it, it becomes very easy · 3 years, 11 months ago

No · 3 years, 11 months ago

Just expand it , and try to put in square form , its very simple · 3 years, 11 months ago

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. Staff · 3 years, 11 months ago

I think, its from Challenge and thrill . right? · 3 years, 11 months ago

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 · 3 years, 11 months ago

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. Staff · 3 years, 11 months ago