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Question

If x1,x2,x3,x4 are four positive real numbers such that x1+1x2=4, x2+1x3=1, x3+1x4=4 and x4+1x1=1, then


A
x1=x3
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B
x2=x4
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C
x1x2=1
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D
x3x4=1
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Solution

The correct options are
A x1=x3
B x2=x4
C x1x2=1
D x3x4=1
Using x+1y2xy, we get4=x1+1x22x1x21=x2+1x32x2x34=x3+1x42x3x41=x4+1x12x4x116=(x1+1x2)(x2+1x3)(x3+1x4)(x4+1x1)24
Hence, each of the above inequality must be an equality.
Therefore, x1=1x2, x2=1x3, x3=1x4 and x4=1x1.x1=2,x2=12,x3=2,x4=12

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