A number $a \in \mathbb{R}$ is an "universal chord" if for every continuous function $f:[0,1] \rightarrow \mathbb{R}$ with $f(0)=f(1)$ exists $x,y \in [0,1]$ with $|x-y|=a$ and $f(x)=f(y)$. Determine all universal ropes.
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1I think you forgot the condition that $|x-y|=a$ – Mike Earnest Feb 25 '15 at 18:32
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http://math.stackexchange.com/search?q=universal+chord – Pedro Feb 25 '15 at 18:43