Apps like the Kindle ones allow you to sync any books you've purchased from them but not for any you get from elsewhere, and as far as I know there are no apps that allow you to sync your own books.
An alternative to this would be to use a web-based ebook reader like Ibis Reader. You upload your books to your account you create with them and then read the book online. Seeing as you are reading it online, it will remember your last page so when you log into it with another device, you can continue from where you left off.
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