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Math Help - contradiction problem

  1. #1
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    Smile contradiction problem

    Very tricky contradiction problem:-

    Basketball player Michael keal's team statistician keeps track of the number, S(N), of the successful free throws he has made in the first N attempt of the season. Early in the season, S(N) was less than 80% of the N, but by the end of the season S(N) was more than 80% of the N. Was there necessarily a moment in between when S(N) was exactly 80% of N?
    Use proof by contradiction and assume there was no such moments. Then S(N) much jump over 80% at one time....
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  2. #2
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    Quote Originally Posted by Vedicmaths View Post
    Very tricky contradiction problem:-

    Basketball player Michael keal's team statistician keeps track of the number, S(N), of the successful free throws he has made in the first N attempt of the season. Early in the season, S(N) was less than 80% of the N, but by the end of the season S(N) was more than 80% of the N. Was there necessarily a moment in between when S(N) was exactly 80% of N?
    Use proof by contradiction and assume there was no such moments. Then S(N) much jump over 80% at one time....
    If there is no such moment then some time in the season there exists a number of free throws N where the number of successes was M and

    \frac{M}{N}<0.8

    but the next free throw was a success and:

    \frac{M+1}{N+1}>0.8

    The first of these inequalities may be rearranges to:

    M-0.8N<0

    and thesecond to:

    M-0.8N>0.8,

    but M-0.8N canot be both less than 0 and greater than 0.8, a contradiction, hence the assumption is false and there must be a point in the season where S(N)=80\%

    RonL
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