Suggestion to a book with lots of number theory problems What I am looking for is a book that contains "infinitely many problems", starts from the 
easiest to high level(that can be found in national and even international olympiads).
Are there such books, because so far I had not found even a one like it.
 A: I liked A course in computational number theory by David Bressoud and Stan Wagon, since I like to use Mathematica.  You probably don't want the otherwise excellent book 104 Number Theory Problems: From the Training of the USA IMO Team
by Titu Andreescu, Dorin Andrica and Zuming Feng because the problems are not graduated.
A: "Problems in Algebraic Number Theory"
http://www.amazon.com/Problems-Algebraic-Number-Graduate-Mathematics/dp/0387221824
And you might consider Marcus "Number Fields"
http://www.amazon.com/Number-Fields-Universitext-Daniel-Marcus/dp/0387902791/ref=sr_1_1?s=books&ie=UTF8&qid=1422568022&sr=1-1&keywords=marcus+number+fields
While it is an (excellent) text, in response to your question, at numerous places in the actual verbiage he asks you to show "why"? 
For contest stuff: Larsen "Problem Solving through Problems"
http://www.amazon.com/Problem-Solving-Through-Problems-Problem-Mathematics/dp/0387961712/ref=sr_1_sc_1?s=books&ie=UTF8&qid=1422568319&sr=1-1-spell&keywords=larsen+problem+solving+through+problems
It does build and you can pick out the topics specific to number theory - but it might be difficult not to be interested in the other material as well.
