已知实数a1 a2 a3.a8,满足a1+a2+.+a8=20,a1a2...a8

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已知实数a1a2a3.a8,满足a1+a2+.+a8=20,a1a2...a8已知实数a1a2a3.a8,满足a1+a2+.+a8=20,a1a2...a8已知实数a1a2a3.a8,满足a1+a2+

已知实数a1 a2 a3.a8,满足a1+a2+.+a8=20,a1a2...a8
已知实数a1 a2 a3.a8,满足a1+a2+.+a8=20,a1a2...a8<12, 求证这八个数中至少有一个小于1

已知实数a1 a2 a3.a8,满足a1+a2+.+a8=20,a1a2...a8
若x≥1,y≥1,则(x-1)(y-1)=xy-x-y+1≥0,
∴xy≥x+y-1.于是
若实数a1,a2,...,a8均不小于1 ,
则a1a2a3a4,a5a6a7a8不小于1,
∴a1a2…a8≥a1a2a3a4+a5a6a7a8-1,
同理a1a2a3a4≥a1a2+a3a4-1,
a5a6a7a8≥a5a6+a7a8-1,
则a1a2…a8≥a1a2a3a4+a5a6a7a8-1≥a1a2+a3a4-1+a5a6+a7a8-1-1≥a1a2+a3a4+a5a6+a7a8-3
同理:
a1a2≥a1+a2-1,
a3a4≥a3+a4-1,
a5a6≥a5+a6-1,
a7a8≥a7+a8-1,
∴a1a2…a8≥a1+a2+a3+a4+a5+a6+a7+a8-7=13,
这与“a1a2...a8