Yes, by contradiction. You suppose that you have won k out of n games with less than 80%, and that the next game you exceed 80%, winning k+1 out of n+1. This gives k/n < 4/5 and (k+1)/(n+1) > 4/5 This gives 5k < 4n and 5k + 5 > 4n + 4 -> 5k + 1 > 4n -> 5k > 4n - 1 but 5k and 4n are integers, yet we have 4n - 1 < 5k < 4n (contradiction). This can obviously be generalized to winning ratios of the form a/(a+1). Other ratios? --Bill Cordwell

This is a problem from the last Putnam exam that had me scribbling (which I don't have in front of me, but no matter)

Say you're shooting baskets and you keep track of your record as you shoot them, for example

0 for 1 1 for 2 1 for 3 1 for 4 1 for 5 2 for 6 etc, you get the idea

Someone specifies a "target ratio" (I think it was 80% in the problem statement) and you're told that during the record of shots, the shooter was below the target ratio, but later was better than the target ratio.

Is there necessarily a moment at which the shooter must have shot exactly that ratio up to that point?

Answer this for 80% and then generalize

-- Thane Plambeck http://www.plambeck.org/ehome.htm

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