Extending on Artem's answer:
The website of the MCCME is probably the current "official" go-to place for this kind of books, both classical and new. See their library, their free-books collections and their book series; as far as I understand, (almost?) all of this is freely downloadable (if you only see an HTML, scroll to its very bottom for a DjVU link). I have no idea whether one of these links is contained in the others.
I am particularly fond of the Populyarnye lekzii po matematike (this is also on the MCCME site, but their DjVU links are currently broken) and the Biblioteka matematicheskogo kruzhka series. The names translate as "Popular lectures in mathematics" and "Library of a mathematical circle", and are meant that way: The popular lectures are written for "normal" interested students, and, e.g., Vorobyov's one on Fibonacci numbers spends a page discussing mathematical induction; the one by Skornyakov gives a very basic introduction into linear algebra; Kaluzhnin proves unique factorization for integers before he ventures into $\mathbb Z\left[i\right]$; etc.. The library of a mathematical circle (starting with the famous Shklyarsky-Chentzov-Yaglom problem book whose first tome was translated into English) is more advanced and tends to be written in a problem-book style; it is, I believe, the kind of problem books where the solutions are written for reading, not just as deadweight. I suspect both of these series would fill gaps in Western education if translated (though a few of them have been already -- and a couple of the Library series are actually translations to begin with).
math.ru, again as far as I can tell, combines the above libraries and some more. Of course, you can get even more books from Library Genesis if you know what to search for.
Among the authors you might recognize Eugene B. Dynkin, Igor Shafarevich, Alexander Gelfond. You probably also have heard of the Yaglom brothers if you are into mathematical olympiads; a number of their books has been translated into English (see the linked Wikipedia pages). I don't currently have texts by Arnold and Gelfand staring at me from the screen but I am pretty sure they wrote some.