(1) The county court or the board of county commissioners of any county and the authorities of any city within the county may enter into an agreement for the construction, improvement or repair of, and the acquisition of right of way for:
(a) Any county road or city street within the corporate limits of the city.
(b) Any road without such city, if it has a population of less than 100,000, but leading directly to it.
(2) In the agreements the parties shall agree upon the proportion which each shall contribute toward such acquisition, construction, improvement or repair and upon the method and kind of acquisition, construction, improvement or repair to be made.