# Splay Trees

Level pending

The number $$\frac{1}{2}$$ is written on a blackboard. For a real number $$c$$ with $$0 < c < 1$$, a $$c$$-splay is an operation in which every number $$x$$ on the board is erased and replaced by the two numbers $$cx$$ and $$1-c(1-x)$$. A splay-sequence $$C = (c_1,c_2,c_3,c_4)$$is an application of a $$c_i$$-splay for $$i=1,2,3,4$$ in that order, and its power is defined by $$P(C) = c_1c_2c_3c_4$$.

Let $$S$$ be the set of splay-sequences which yield the numbers $$\frac{1}{257}, \frac{2}{257}, \dots, \frac{256}{256}$$ on the blackboard in some order. If $$\sum_{C \in S} P(C) = \tfrac mn$$ for relatively prime positive integers $$m$$and $$n$$, compute $$100m+n$$.

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