I know that this question may be weird or ridiculous to some but— Can anyone suggest me a book which can effectively bridge the gap between High School and the Olympiads?
I have done "Higher Algebra by Hall and Knight" and doing "Higher Algebra by Bernard and Child". Both of these books are really good and give clear cut theory and start with the basic problems all the way to the difficult ones (for me) thought they lack solved examples.
I want a book like the above two which would cover all the areas of mathematics required for Olympiads, give all the required theory in short with proofs and has decent amount of solved and unsolved problems ranging from basic to the Olympiad level.
Also, try not to recommend me the book which would present basic and obvious solved examples and then proceed to give the most difficult problems it would find. In short, I wouldn't prefer a book in which the difference in the difficulty between the examples and problems is too much.
Essentially, I want a book which would start with basic theory and basic problems and take them all the way to the Olympiad level.
For a couple of decades Problem-Solving Strategies by Arthur Engel has been one of the most highly recommended books.
Lesser known, but perhaps also very appropriate for what you want (because they have lots of short "theory" sections motivated by some of the problems), is the 2-volume Hungarian Problem Book I and Hungarian Problem Book II. Despite dating from 1894-1905 (I) and 1906-1928 (II), the problems and theory discussions are still very appropriate for present-day contests.
Over 100 books, along with informative reviews by mathematics professionals (whatever that means), can be found in MAA's book list under the category Problems Olympiad Level. Also, there are the books listed at Art of Problem Solving (which seems more limited than it used to be, but maybe my memory is off).