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Omni-MATH / For any two different real numbers x and y, we define D(x,y) to be the unique integer d satisfying 2^(d)≤|x−y|<2^(d+1). Given a set of reals mathcal F, and an element xin mathcal F…

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problem

For any two different real numbers xx and yy, we define D(x,y)D(x,y) to be the unique integer dd satisfying 2d≤|x−y|<2d+12^d\le |x-y| < 2^{d+1}. Given a set of reals $\mathcal F$, and an element $x\in \mathcal F$, we say that the [i]scales[/i] of xx in $\mathcal F$ are the values of D(x,y)D(x,y) for $y\in\mathcal F$ with x≠yx\neq y. Let kk be a given positive integer. Suppose that each member xx of $\mathcal F$ has at most kk different scales in $\mathcal F$ (note that these scales may depend on xx). What is the maximum possible size of $\mathcal F$?
Plain-text mathematical notation (without MathML)
For any two different real numbers x and y, we define D(x,y) to be the unique integer d satisfying 2^(d)≤|x−y|<2^(d+1). Given a set of reals $\mathcal F$, and an element $x\in \mathcal F$, we say that the [i]scales[/i] of x in $\mathcal F$ are the values of D(x,y) for $y\in\mathcal F$ with x≠y. Let k be a given positive integer. 
 
Suppose that each member x of $\mathcal F$ has at most k different scales in $\mathcal F$ (note that these scales may depend on x). What is the maximum possible size of $\mathcal F$?
Original LaTeX notation
For any two different real numbers $x$ and $y$, we define $D(x,y)$ to be the unique integer $d$ satisfying $2^d\le |x-y| < 2^{d+1}$. Given a set of reals $\mathcal F$, and an element $x\in \mathcal F$, we say that the [i]scales[/i] of $x$ in $\mathcal F$ are the values of $D(x,y)$ for $y\in\mathcal F$ with $x\neq y$. Let $k$ be a given positive integer. 
 
Suppose that each member $x$ of $\mathcal F$ has at most $k$ different scales in $\mathcal F$ (note that these scales may depend on $x$). What is the maximum possible size of $\mathcal F$?

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Source and history

task_id
62650104-ece1-56e4-b045-8f454331ead6
task_key
test--62650104-ece1-56e4-b045-8f454331ead6
task_revision_id
1
config / split
/ test

Official source

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