Do I need to write something like ‘sincerely’ or my name/signature on an essay which I have to attach to my documents when I am applying for internship to U.S? This essay should answer the question what are my expectations from this internship and how participating in this internship can benefit my life. My problem is that I would like to fit with the text I wrote on one page and if I put some more lines (2-3) I will need another page. There is no way to make the essay shorter – I tried and I have the minimum I would like to say. Please, give me any advices, I would really appreciate that.

Trouble Maker Emotion: wink
If it is an essay, then you don't sign it.

However, these applications usually ask for a motivation letter detailing the things you mention, and if so, you should treat it as a letter.

I'm sure you can cut a couple of lines if you really try. Otherwise try making the text size half a point smaller? Or amending the margin sizes?
Thank you for your answer Nona. I really appreciate that