Which journals publish research announcements? Can anybody advise mathematical journals that publish research announcements? (I mean little papers without proofs.) 
It sometimes happens that a proof is so long that it takes years to review and few journals are willing to accept it. I think in this case it would be useful (apart from posting your paper in arXiv) to  announce the result in a short communication (this is how this problem is resolved in Russia, but I am asking about the rest of the world).
I am working in Functional analysis and in Geometry.
EDIT. More generally, I am curious, 

how do people solve the problem of long texts? 

Suppose you write a long text, where the main results are just several propositions, say, 5 theorems, and the rest are various technical lemmas, many of them, say, 100. Your text is devoted to the explanation of one idea, those 5 theorems appear only in the end, and it is impossible to separate them so that some of them could be proved in the middle of the text. As a corollary, it is impossible to divide your text into several papers, on 40-50 pages, so that they could be sent to usual journals.
You have to send this long paper to a journal, that publishes long texts, they will spend some time on finding a reviwer, he will check everything in your text, after that they will put your paper into the queue, and this is a long process, you understand... It can take 2-3-5 years before your paper will be published.
Of course, it will be difficult to explain to your employer, what you were doing the prevoius 5 years, when you were writing this paper. 
So how do people resolve this problem? In Russia there is a possibility to write several  little papers, without proofs, but with the formulations of the main results (those 5 theorems), then send them to some journals (together with that long text with accurate proofs), then the reviewer checks everything, agrees, they publish these little papers (this is much quicker, since these papers are little, there is no need to put them into a long queue), and everything is OK. As far as I understand, in France the situation is more or less similar. What about the rest of the world?   
 A: Tl;dr: I don't think the math publication ecosystem has a perfect solution to your problem.
First, note that even moderate-length papers can take years to be published (my personal record is of 5 years from submission to print, involving only one submission, for a 15-pages paper). If your employer only considers published papers, and not accepted or arXiv posted ones, you are basically out of good options (of course if any journal where to count, you could find a quick predatory journal, but I would not advise this).
The usual process is to post the complete paper on the arXiv and submit to a suitable journal (Memoires de la SMF is another good available option, similar to Memoirs AMS, and I think Documenta is able to publish long papers). Publishing an announcement has become rare, but is still possible (several venues for this have been mentioned already). However, when you say that proofs are thoroughly check, that is usually not true; and if it were, then you announcement would also take much time to be accepted, since the review time would be long.
So, it somehow seems that your main concern is about the accepted-to-published delay. I thus advise you to have a look at the AMS data on delays (published yearly in the Notices), you might find a good venue. Also, you could have a shot at an electronic only journal where this delay can be much shorter. Forum of Mathematics (Sigma or Pi) could be an option if your paper is very strong, but I think they now charge authors, and you will have the usual problem that you could spend a year or more waiting for a report which could turn out negative.
We lack a math megajournal which would publish any good paper (good meaning good enough to deserve publication) electronically, avoiding unecesary delays in resubmission along the prestige ladder and in finding room into issues of fixed length. That would be about the best option for you, where it to exist.
To finish on another note: once your large paper is written and waiting for a venue, you may consider working on smaller scale projects to get your beans to be counted. I do not like to advise this, but given the incentives you mention that might be the only solution. It is a bit like a painter working for hire in advertisement and buying himself time for his masterpiece; be careful that your smaller scale project have some interest instead of being noise in the publication record.
A: In addition to the above list let me add Comptes rendus, Proceeding of the National Academy,
Electonic research announcements, Bulletin of the London Math. Soc.
In addition to this there are various conference proceedings.
EDIT. Before you edited the question I thought that your problem is with publishing a short paper. But now I see that your problem is opposite :-)  Of the journals mentioned in the answers, Math Notes and Bull LMS do not publish announcements without proofs. A 70-100 pages
paper indeed causes problem, not speaking of the paper with 100 lemmas and propositions. I think that a paper with 100 lemmas is a book, and there are very few journals which wuld consider this. 
To convince your employer that you were not idle, post this enormous paper on the arxiv.
Now I do not understand your concern about the country where you work: I think Comptes rendus and other journals mentioned, including Russian DAN, and Funk Analiz, publish papers from all countries.
A: Look at Mathematical Notes and
Functional Analysis and its Applications and Russian Mathematical Surveys and Doklady RAN and Proceedings of the AMS.
