Let x, y, z be complex numbers such that aligned & (x)/(y+z)+(y)/(z+x)+(z)/(x+y)=9 & (x^2)/(y+z)+(y^2)/(z+x)+(z^2)/(x+y)=64 & (x^3)/(y+z)+(y^3)/(z+x)+(z^3)/(x+y)=488 aligned If (x)/(y z)+(y)/(z x)+(z)/(x y)=(m)/(n)

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Correct answer: 16

aligned & (x)/(y+z)+(y)/(z+x)+(z)/(x+y)=9 & (x+y+z)((1)/(y+z)+(1)/(z+x)+(1)/(x+y))=12 aligned Let x+y+z=S_1 (1)/(y+z)+(1)/(z+x)+(1)/(x+y)=(12)/(S_1) Now, (x+y+z)((x)/(y+z)+(y)/(z+x)+(z)/(x+y))=9 S_1 aligned & 64+S_1=9 S_1 & S_1=8 & (x+y+z)((x^2)/(y+z)+(y^2)/(z+x)+(z^2)/(x+y))=64 S_1 & 488+(x^2+y^2+z^2)=64 × 8 & x^2+y^2+z^2=24 & x y+y z+z x=(64-24)/(2)=20 aligned aligned & Now, (1)/(x+y)+(1)/(y+z)+(1)/(z+x)=(12)/(8)=(3)/(2) & (1)/(8-x)+(1)/(8-y)+(1)/(8-z)=(3)/(2) & (8-x)(8-y)(8-z)=56 & x y z=104 aligned So, (x)/(y z)+(y)/(z x)+(z)/(x y)=(24)/(104)=(3)/(13)=(m)/(n) m+n=16

Let x, y, z be complex numbers such that aligned &...

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