I am looking to build a repertoire of olympiad type problems which have non-intuitive elegant solutions, If possible instead of a resource, I think problems would be the best. (i.e. select the best problem to post here).
The problem I like best that falls into this category is to prove that if a bigger rectangle has some smaller rectangles completely space filling inside it, and the small rectangles have at least one side of integer length, then we need to show that the big rectangle has at least one integer length.
The non-intuitive(to me) solution is to place each smaller rectangle on a checkerboard with side length of the pattern = 1/2. Then to note that each smaller rectangle must have an equal area of black and white. Then to prove that for any checkerboard to have an equal area of black and white, it must have one of the lengths of integer length.