MacBook Pro is not turning on monitor When I turn on my MacBook Pro it starts with a sound of checking CD inside and the case light detector turns on as usual, and ventilators starts to run, but monitor is still switched off.

At the beginning right after it happened, I just switched off/on the computer once and it started again with all it's processes and a typical beginning sound, but without a screen. Moreover, I was able even to login (in blind mode) and really check if the machine is running, because my Internet sharing has been started as soon as Mac OS starts. So my network has been accessible in other computers and I was sure that the problem has been just in my monitor.

Than I tried to turn the computer on/off manually by holding power button many, many times. And it almost always did the same process, starting with a typical Mac sound at the beginning. But later on, it stopped to make this sound, however turned on with a sound of checking the CD, running ventilators. But I am not able to login anymore and seems login view is not loaded anymore. Monitor is still not turning on at all. Perhaps, this could happen, because I was pushing on/off button too many times or so?!

But right now, I am having it turning on, just with ventilators, case light and a sound of checking the CD, and that's it. Computer is opened and located here on my table ready for any checks...

I already has been in Apple center and they made a 3 days diagnostic, saying my motherboard is broken, but I don't believe them at all. Services are made very bad here in Austria. I was trying to explain that I was able even to login, but the conclusion has been final - motherboard and -$50.

So, I don't trust in their checking at all and wish to hear your opinions, if this is motherboard or some other stuff could be broken?!
It could very well be a broken logic board.  But try to do the following and see if it fixes it :

Reset System Management Controller as per these instructions

Reset PVRAM as per these instructions

Doing these two resets generally fixes most hardware based issues on Macs.  You can also eliminate that its your monitor by trying to see if you can replicate this problem on other screens.

But I would trust the advice you've been given by Apple on this one.