Chris wrote: << . . . << Thane Plambeck asked Say you're shooting baskets and you keep track of your record as you shoot them . . . 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?

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so i now think that the answer is yes for 80% = 4/5 and any other a/(a+1) fraction, but no for any others

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I found the answer to this question (I think from the last Putnam exam) to be totally counterintuitive. (The target ratio was in fact 80%.) Here's the proof from the Monthly: As you come up from below 80% to greater than it, there is a last moment when you're still below 80%. Say that was when you'd made p-1 baskets out of q-1 tries; after the next shot you're greater than or equal to 80%. Therefore: p-1/q-1 < 4/5, implying 4q+1 > 5. And 4/5 <= p/q, implying 4q <= 5p The two inequalities guarantee that 4q = 5p. Amazing. --Dan