### IMO Shortlist 2013 problem C3

Kvaliteta:

Avg: 4,0Težina:

Avg: 7,0 A crazy physicist discovered a new kind of particle which he called an

Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.

*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 of each imon . During this procedure, the two copies and become entangled if and only if the original imons and are entangeld, and each copy becomes entangled with its original imon ; 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.

Izvor: Japan