2013 IMO Shortlist C3

A crazy physicist discovered a new kind of particle which he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time. (i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it. (ii) At any moment, he may double the whole family of imons in his lab by creating a copy $I^{\prime}$ of each imon $I$. During this procedure, the two copies $I^{\prime}$ and $J^{\prime}$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I^{\prime}$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment. Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled. (Japan)

一位疯狂的物理学家发现了一种新的粒子，他将其称为“离子”，其中一些粒子神秘地出现在他的实验室中。实验室中的一些离子对可以纠缠，并且每个离子可以参与许多纠缠关系。物理学家找到了一种用这些粒子执行以下两种操作的方法，一次一种操作。 (i) 如果一些离子在实验室中与奇数个其他离子纠缠在一起，那么物理学家就可以摧毁它。 (ii) 在任何时候，他都可以通过创建每个 imon $I$ 的副本 $I^{\prime}$ 将其实验室中的整个 imon 家族加倍。在此过程中，两个副本 $I^{\prime}$ 和 $J^{\prime}$ 变得纠缠当且仅当原始 imon $I$ 和 $J$ 纠缠时，并且每个副本 $I^{\prime}$ 与其原始 imon $I$ 纠缠；此刻没有其他纠缠发生或消失。证明物理学家可以应用一系列这样的操作来产生一个伊蒙族，其中没有两个是纠缠的。 (日本)