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Demonstrate the inequality: [(a+b)/c]*[(b+c)/d]*[(c+d)/a]*[(d+a)/b]>16

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Demonstrate the inequality:

[(a+b)/c]*[(b+c)/d]*[(c+d)/a]*[(d+a)/b]>16

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We know that (sqrt a + sqrt b)^2=a+b+2sqrt (a*b)

a+b>2sqrt (a*b) and (a+b)/c>2sqrt (a*b)/c

c+b>2sqrt (c*b) and (c+b)/d>2sqrt (c*b)/d

c+d>2sqrt (c*d) and (c+d)/a>2sqrt (c*d)/a

a+d>2sqrt (a*d) and (a+d)/b>2sqrt (a*d)/b

(a+b)/c*(c+b)/d*(c+d)/a*(a+d)/b>16sqrt(abcd)^2/abcd=16

q.e.d.

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