2010 Putnam B3

There are 2010 boxes labeled $B_1, B_2, \dots, B_{2010}$, and $2010n$ balls have been distributed

among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves,

each of which consists of choosing an $i$ and moving *exactly* $i$ balls from box $B_i$ into any

one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls

in each box, regardless of the initial distribution of balls?

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已分发 2010 个标有 $B_1、B_2、\dots、B_{2010}$ 的盒子和 $2010n$ 球

其中，对于某个正整数$n$。您可以通过一系列动作重新分配球，

每个步骤都包括选择一个 $i$ 并*准确地*将 $i$ 球从 $B_i$ 盒子移动到任何一个

另一个盒子。对于哪些 $n$ 值，可以用恰好 $n$ 个球来达到分布

在每个盒子里，无论球的初始分布如何？

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