Search text in cyrillic alphabet [duplicate]

I have problem with documents generated in LaTeX with Cyrillic symbols. I created the following file:

\documentclass{article}\usepackage[utf8]{inputenc}\usepackage[T2A]{fontenc}\begin{document}ЛаТеХ\\LaTeX\end{document}

And when I copy text in Cyrillic alphabet, I obtain some gibberish on the clipboard,

This makes my articles quite hard to quote. So my first question is, can I fix this and have proper copying process?

Also if I anyway type certain text and look for it in the document, it is not found in the document, being absolutely clear, that this text is there:

My second question is, can I make the document in such a way, that I can obtain returned the Cyrillic string, that I seek?

All this doesn't happen, if the letters are Latin. The Cyrillic letters, however, cause these problems and at this point I cannot use LaTeX due to these restrictions.

I will be happy to give further details, when needed.

PS: I use "MikTex", pdfLatex+MakeIndex+BibTex.