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Free Throw Shooting

The Problem

Shaq starts the season shooting below 50% from the free throw line. He finishes the season shooting above 80%. Was there necessarily a moment when his free throw percentage was exactly 75%?

What Makes This Problem Non-Trivial

Let’s say we asked whether there was ever a point where the FT% was 60%. Consider this sequence of events.

MathJax TeX Test Page $$\dfrac{0}{1} \to \dfrac{1}{2} \to \dfrac{2}{3} \to \dfrac{2}{4} \to \dfrac{2}{5} \to \dfrac{3}{6} \to \dfrac{4}{7} \to \dfrac{5}{8} = 0.625$$ There was never a point where it was 60%. It went from $57.14\% \to 62.5\%$. Are we guaranteed that this doesn't happen for $75\%$?

The Solution

MathJax TeX Test Page

Boy/Girl Population