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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 where m, n are positive integers with GCD (m, n)=1 , find m+n .

Correct answer

16

Step-by-step solution

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=S₁ 1 y+z + 1 z+x + 1 x+y = 12 S₁ Now, (x+y+z) ( x y+z + y z+x + z x+y )=9 S₁ aligned & 64+S₁=9 S₁ & S₁=8 & (x+y+z) ( x^2 y+z + y^2 z+x + z^2 x+y )=64 S₁ & 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

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