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Problem 4.
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
Solution
The answer is $C= \frac{1}{2}$

For $C= \frac{1}{2}$, consider the following strategy for Turbo. Choose an arbitrary point $X$ (which is not Turbo's starting point), and always jump in the direction of $Y$ which is the antipode of $X$. Indeed, this strategy works by induction.

Now, assume $C > \frac{1}{2}$. Consider the following construction to guarantee every point is covered. First, start off the sequence with \[\frac{1}{2}, \frac{1}{4}, \frac{1}{2}\]This will guarantee that turbo will have crawled across every point, except some interval $T_0$ of length $\frac{1}{4}$, and that turbo will be standing right on an endpoint of $T_0$. Now, let the next number in the sequence be the distance from any endpoint of $T_0$ to the antipode of the midpoint of $T_0$. This forces Turbo to the antipode of the midpoint of $T_0$, otherwise, he must cover $T_0$ and, by extension, the whole circle. From here, choose some $\frac{1}{2} < \alpha < C$ and make turbo move a distance of $\alpha$. This reduces the size of the interval of uncovered points by exactly $\alpha-\frac{1}{2}$. Let $T_1$ be the new set of uncovered points. Then, do the exact same process we did on $T_0$ to $T_1$. In general we have \[ \text{length of }T_k = \frac{1}{4} - k \left ( \alpha - \frac{1}{2} \right ) \]which is negative for sufficiently large $k$.
 
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