1995 Putnam A1

Let $S$ be a set of real numbers which is closed under

multiplication (that is, if $a$ and $b$ are in $S$, then so is $ab$).

Let $T$ and $U$ be disjoint subsets of $S$ whose union is $S$. Given

that the product of any {\em three} (not necessarily distinct)

elements of $T$ is in $T$ and that the product of any three elements

of $U$ is in $U$, show that at least one of the two subsets $T,U$ is

closed under multiplication.

令 $S$ 为一组实数，其封闭于

乘法(也就是说，如果 $a$ 和 $b$ 在 $S$ 中，那么 $ab$ 也在 $S$ 中)。

令 $T$ 和 $U$ 为 $S$ 的不相交子集，其并集为 $S$。给定

任何{\em三个}的乘积(不一定不同)

$T$ 的元素在 $T$ 中，并且任意三个元素的乘积

$U$ 位于 $U$ 中，证明两个子集 $T,U$ 中至少有一个是

乘法下封闭。