How do you say phrases of encouragement such as "good luck" or "do your best!" or "you can do it!" in Russia?
Hello! Thanks for the question! I’ve moved it to the top of my priority list since it sounds like you may require the answer ASAP.
“Good luck!” in Russian is “Желаю удачи!” (Zhelayu udachi (I’ll be bolding the stressed vowels from now on btw)) or just “Удачи!”. I wouldn’t advise telling a Russian to "do their best”, ‘cause to us it sounds like an implication that one wouldn’t do their best unless you told them to, lol. “You can do it!” is also not very often used, but you can translate it as “Ты сможешь!” (Ty smozhesh) or “Ты сделаешь это!” (Ty zdelayesh eto).
So yeah, the safest bet is to wish them luck! There’s one more “pro” way to do it - you can say “Ни пуха ни пера!” (Ni puha ni pera), to which you can expect an answer of “К чёрту!” (K chyortu). Literally it translates to “[Wishing you] no fur, no feather!” - “[Go] to the devil!”. I swear I’m not setting you up for a faux pas xD My family and I had this exchange before every single one of my exams. Apparently this expression originated from hunters wanting to wish each other good luck while trying not to jinx it, so the workaround was to wish the opposite of luck in hunting: to get no fur (beast) nor feather (bird). I didn’t know this until now! :P
Hope this was useful, and wishing luck to whoever you’re going to wish it to!