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OlympiadBench / 1920 / For any two different real numbers x and y, we define D(x,y) to be the…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

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} \leqslant|x-y|<2^{d+1}. Given a set of reals F\mathcal{F}, and an element x∈Fx \in \mathcal{F}, we say that the scales of xx in F\mathcal{F} are the values of D(x,y)D(x, y) for y∈Fy \in \mathcal{F} with x≠yx \neq y. Let kk be a given positive integer. Suppose that each member xx of F\mathcal{F} has at most kk different scales in F\mathcal{F} (note that these scales may depend on xx ). What is the maximum possible size of F\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 F, and an element x∈F, we say that the scales of x in F are the values of D(x,y) for y∈F with x≠y.

Let k be a given positive integer. Suppose that each member x of F has at most k different scales in F (note that these scales may depend on x ). What is the maximum possible size of 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 $2^{d} \leqslant|x-y|<2^{d+1}$. Given a set of reals $\mathcal{F}$, and an element $x \in \mathcal{F}$, we say that the scales 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 $\mathcal{F}$ ?

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID
1920
task_id
0c045671-fbe8-5ee7-8b03-ed87a8f43027
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1920
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import