However, according to wikipedia:
plus one of the top five individual scorers (designated as Putnam Fellows) is awarded a scholarship of up to $12,000 plus tuition at Harvard University (Putnam Fellow Prize Fellowship),1
William Lowell Putnam Mathematical Competition
Then the exam thing came around, and one of the prizes was that one of the winners, one of the five, would be chosen for a scholarship to Harvard for the graduate school — I believe, if I remember right. The method, as I understood it, was they didn’t want to take the winner, because they didn’t want to be stuck with a nut of some sort, by accident, right? So they made it that out of the first five they could choose, Ok?
Interview of Richard Feynman by Charles Weiner on 1966 March 5, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA.
So from the same interview, he mentioned that the math contest is national and the top fives are awarded Harvard scholarship. I am leaning towards Putnam. :)