Justatheory… how much do you know about telescopes?
When people talk about powerful they mean in the context of magnification. In astronomy we are more interested in light gathering power which means larger aperture (more light can enter the telescope to form an image). Due to turbulent air currents (caused by ground heating from the Sun) telescopes tend to perform quite poorly during the day over land (or water in this case).
The effect of turbulence degrades the quality of the image you will get over relatively short distances of a few miles even. This seriously limits the resolving power of telescopes. According to Google Maps, the only immediately available source of information I have, the distance between NZ and Argentina is a little over 4,600 miles (I get a different figure each time I measure). Over that distance, even if dead flat, it would have to be a really big sign for you to be able to make anything out.
As a theory or mind experiment then yes it would work but in practice, no way that would work. The Moon is a little over a quarter of a million miles away and you can see that with your naked eye because of its size. The smallest craters you can see with a good telescope and stable seeing conditions are a mile or so across. So that gives you some guidance on how big your sign would have to be to see it through a telescope assuming you have direct line of sight. Over 4,600 miles distance over land you wouldn't have that so experiment wouldn't work.