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[SOLVED] Inequality involving a, b, c and d

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anemone

MHB POTW Director
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Feb 14, 2012
3,909
Given the real numbers $a,\,b,\,c$ and $d$, prove that

$(1+ab)^2+(1+cd)^2+a^2c^2+b^2d^2\ge 1$
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,909
Expanding the LHS of the inequality we get $1+2ab+a^2b^2+1+2cd+c^2d^2+a^2c^2+b^2d^2=1+(1+ab+cd)^2+(ac-bd)^2\ge 1$