She has a 50 point lead, and on the final fifty shots, the person in second place can score at most 500 points (10 points per shot). So as long as she scores more than 450 points on her final fifty shots, she will be certain to win.

If she gets n consecutive bullseyes, scoring 10 points each, she will get 10n points in total. Then on her final 50-n shots, we know she also scores at least 4 points, so she will get at least another 4(50 - n) = 200 - 4n points. So she will score at least 10n + 200 - 4n = 6n + 200 points. This needs to be greater than 450, so

6n + 200 > 450
6n > 250
n > 41 + 2/3

and the smallest possible integer value of n is 42.
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My solution
Max points that can be lost on each shot is 6 points. Lead is 50. So for 8 shots they can afford to not hit bulls eye while opponents hit bullseye. So 42 guarantees victory

Posted from my mobile device
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