Stat Tracking

A mathlete has been tracking the percentage of problems he has gotten right so far this season.

Let nn be the number of problems given to him, and mm the number he has gotten correct.

Find the sum of all 0<q<0.850<q<0.85 such that if at some point in the season mn<q\frac{m}{n}<q, and at some later point in the season mn>q\frac{m}{n}>q, then there will always be a third point in the season where mn=q.\frac{m}{n}=q.

If you believe that no such qq exists, submit 1-1 as your answer.

Clarification: Although it is true that any rational qq may be obtained after the first two points, the problem requires that this rational qq must exist at some point in the season. In particular, it might be helpful to analyze points in the season between the two points satisfying the first two conditions.

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