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[不等式] 8 个正数都大于等于 1,它们的和为 20,那么它们的积最小能小到多少呢?

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TSC999 posted 2023-12-9 20:31 |Read mode
如果 8 个正数都大于等于 1,它们的和为 20,要使它们的积尽可能的小,最小能小到多少呢?

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kuing posted 2023-12-9 21:54
若有两个数 `>1`,不妨设 `a>1`, `b>1`,则令 `a'=1` 和 `b'=a+b-1`,有 `a'+b'=a+b` 且 `a'b'-ab=-(a-1)(b-1)<0`,所以将 `a`, `b` 变成 `a'`, `b'` 可使和不变而乘积更小。
因此对于原题,取最小时必定 7 个为 1,另一个为 13。

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