Should I admit that I can't find a solution to a math problem? The problem is pretty much the title. I have a bachelor and master degree in applied math. I have been assigned a small problem to start with, from a professor (who is considered very good in my field) who is willing to give me a reference, which in turn I need in order to apply for a PhD position. Our common goal is for me to work gradually on the problem and, if I solve it, produce a published paper. This is a huge matter for me as do not yet have any publications, and it will (I believe) boost my chances to obtain a PhD position. The problem is that I have been working for a month on this and I have not reached an answer. So, considering that he said that this first problem would be pretty easy to deal with, I am really wondering what to do.

I need an answer on this: Should I admit to this person that is highly accomplished that I am having difficulties at what he asked from me? or should I tell him all I have got up till now even if that is not even close to a solution?
I don't know why you think there are any options other than both.  Of course you shouldn't just say "I'm having difficulties, full stop" without talking about what you have tried.  Of course you shouldn't just say "this is what I did" without also talking about where you're stuck.

You sound like you're worried about making the best impression on this professor.  The best impression you can make here is showing that you have the maturity to deal with doing research: to work hard on a problem, be aware of both what you've accomplished and what you haven't, and be able to ask for help where you need it.

There are bunch of things that could be happening:


You went off in the wrong direction, or missed something, and the professor will be able to help get you back on the right track.
The problem is actually harder than the professor thought, and explaining why you're stuck will make that clear to them.
The problem is harder than you thought, and the professor expected you to need some help.


In all of those cases, the absolute best thing to do is honestly present the work you've done.  The less you worry about framing things to make you look good, the better you actually end up looking.