1995 Putnam B1

For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$,

let $\pi(x)$ be the number of elements in the part containing $x$.

Prove that for any two partitions $\pi$ and $\pi'$, there are two

distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$

such that $\pi(x) = \pi(y)$ and $\pi'(x) = \pi'(y)$. [A {\em

partition} of a set $S$ is a collection of disjoint subsets (parts)

whose union is $S$.]

对于 $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ 的分区 $\pi$，

令 $\pi(x)$ 为包含 $x$ 的部分中的元素数量。

证明对于任意两个分区 $\pi$ 和 $\pi'$，有两个

$\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ 中不同的数字 $x$ 和 $y$

这样 $\pi(x) = \pi(y)$ 和 $\pi'(x) = \pi'(y)$。 [一个{\em

集合$S$的分区}是不相交子集(部分)的集合

其联合是 $S$。]