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答：由于对称性，不妨设2≤x≤y≤z.所以xy≤yz,xz≤yz.1/x+1/y+1/z=1.两边都乘以xyz,yz+xz+xy=xyz.所以xyz=yz+xz+xy≤yz+yz+yz=3yz,所以x≤3.由2≤x≤3.当x=2.1/y+1/z=1/2.2(z+y)=yz.(y-2)(z-2)=4.所以y-2=1,z-2=4或y-2=2,z-2=2或y-2=4,z-2=1于是得到解(x,y,z):(2,3,6),(2,4,4).当x=3.1/y+1/z=2/3.3(z+y)=2yz.于是(2y-3)(2z-3)=9.所以2y-3=1,2z-3=9或2y-3=3,2z-3=3或2y-3=9,2z-3=1于是得到解(x,y,z):(3,3,3).综上，原方程的解为(x,y,z):(3,3,3)(2,4,4)(4,2,4)(4,4,2)(2,3,6)(2,6,3)(3,2,6)(3,6,2)(6,2,3)(6,3,2) |
