Wow, this is really motivating. I just started learning Japanese about a month ago with one of my friends, and now we've expanded into a group of six. I'm going to print this out and put it in my notebook.
Just wanted to let you know that the word you have for "stay" actually means like..to reside somewhere, not so much the same meaning as the English in this case. I can't think of a fitting way to say "stay strong" in Japanese but you might want to try to find a japanese person to help you out?
If not then even something as simple as "がんばって(ganbatte)" I feel would get the feeling across, as far as words that people say to encourage each other.
Yeah, I don't really know anyone that knows japanese so I had to go with Google Translate xD I had a feeling it was gonna be wrong! If I'll update it sometime I'll be sure to look some words up, but I'll leave it like this on this one