好难已知a×b×c≠0,x、y、z为整数,且x+y+z=0,a∧yz=b∧xz=c∧xy,求a×b×c的值.

Question

好难
已知a×b×c≠0,x、y、z为整数,且x+y+z=0,
a∧yz=b∧xz=c∧xy,求a×b×c的值.

Answer 1

不妨设a^yz=b^xz=c^xy=A,
则a=A^(1/yz)
b=A^(1/xz)
c=A^(1/xy)
则abc=A^(1/yz+1/xz+1/xy)
=A^[(x+y+z)/xyz]
=A^0
=1