Exposed to radiation doesn't always equal being radioactive

Factual Questions
1

So, a quick review of relevant GQ threads shows that pure water can become radioactive (for a few days) after being exposed to radiation. A search of the web shows that some elements can remain radioactive for a significantly longer time (e.g. Cobalt 60 has a half-life of 5.27 years).

Yet on a recent mythbusters segment (will cockroaches survive a nuclear blast?) Grant said “… just because they were exposed to radiation, doesn’t mean they’re radioactive”. He say this while handling an array of ~1" plastic cages each holding a single test insect.

So, why aren’t those radioactive? If simple water can be radioactive for a couple of days why wouldn’t the larger molecules in plastic also become radioactive? (I’m not considering the bugs as even the simplest living organisms are incredibly complex at the molecular level, and also consist primarily of elements that are hard to activate)

Also, why can water become radioactive when hydrogen and oxygen are both considered difficult to activate? Is it simply that difficult does not mean impossible, or is there something about the combination of the two that makes them more susceptible to (neutron) activation?

2

There are different kinds of radiation, and not all of them induce radioactivity in exposed materials.

The article specifically mentions Cobalt-60, which is what’s used to irradiate food, and presumably the roaches on Mythbusters.

3

It also supplies an answer to one of the OP’s questions:

4

Thanks. That link mentioned photodisintegration, a 2nd form of induced radioactivity that’s far more common the the neutron activation I’d read about.

Thus, the answers to all my questions seem to be related to the difference between photodisintegration and neutron activation.