Apologies for the crass title. However, there are incentives for many children in grades (comparable to the U.S.) 3-12 to learn to do arithmetic well. (I recall Number Sense and related contests from my school days.) I would like to mention to students I am tutoring of current programs that offer such rewards. There are of course the science competitions of Westinghouse, Intel, Google, and such, and competitions like Putnam and (used to be called) AHSME, but I am looking for more, especially offered at a regional or local level.
(I choose arithmetic as an example, but I am also interested in such things for algebra, geometry, and related areas. Further, for this question I want to disallow coding challenges and similar programs that end up requiring more than basic mathematics.)
(Added 2015.01.27: I appreciate Joseph O'Rourke's take on the question. I think that "joining an elite group" is differently persuasive from "getting something for demonstration of ability". Also, while I think for some tournaments and competitions can suffice for motivation, in this question I am interested in something like money, i.e. folding stuff, or scholarships, or texts, or gift cards, however that might be. In my case, participating in number sense contests got me into other branches of science, other contests, and acquiring medals, ribbons, trophies, a chemical engineering handbook, some small scholarships, and thanks to AP tests, saving a year of tuition by placing out of many introductory college courses. It also turned into a salary bump in my first job. I would like to say to a student that there are many benefits to learning how to do arithmetic well, and that some of them turn into cash equivalents, not just things that will help them live better lives.
I am hoping there are merit-based scholarships, local foundations that support students based on ability, programs that give rewards and other incentives for supporting mastery of basic skills. I can name several such programs today for reading and science for elementary school and beyond. I am having trouble naming them for mastering skills such as algebra, geometry, and basic arithmetic that are not also general science competitions. Thus this question: what are such math-focused programs? I hope this makes the question more clear. End Added 2015.01.27)
Gerhard "Scholarships Also Count As Rewards" Paseman, 2015.01.26