Putnam 2004, Problem A-1

Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N)$, of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than $80\%$ of $N$, but by the end of the season, $S(N)$ was more than $80\%$ of $N$. Was there necessarily a moment in between when $S(N)$ was exactly $80\%$ of $N$?

Solution:
Yes. Suppose otherwise. Then there would be an $N$ such that $S(N) < .8N$ and $S(N+1) > .8(N+1)$; that is, O'Keal's free throw percentage is under 80% at some point, and after one subsequent free throw (necessarily made), her percentage is over 80%. If she makes $m$ of her first $N$ free throws, then $m/N < 4/5$ and $(m+1)/(N+1) > 4/5$. This means that $5m < 4N < 5m+1$, which is impossible since then $4N$ is an integer between the consecutive integers $5m$ and $5m+1$.

**Remark:** This same argument works for any fraction of the form $(n-1)/n$ for some integer $n > 1$, but not for any other real number between 0 and 1.