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Omni-MATH / Let S be a set of positive integers satisfying the following two conditions: - For each positive integer n, at least one of n,2n,…,100n is in S. - If a₁,a₂,b₁,b₂ are positive integ…

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problem

Let SS be a set of positive integers satisfying the following two conditions: - For each positive integer nn, at least one of n,2n,…,100nn, 2 n, \ldots, 100 n is in SS. - If a1,a2,b1,b2a_{1}, a_{2}, b_{1}, b_{2} are positive integers such that gcd(a1a2,b1b2)=1\operatorname{gcd}\left(a_{1} a_{2}, b_{1} b_{2}\right)=1 and a1b1,a2b2∈Sa_{1} b_{1}, a_{2} b_{2} \in S, then a2b1,a1b2∈Sa_{2} b_{1}, a_{1} b_{2} \in S Suppose that SS has natural density rr. Compute the minimum possible value of ⌊105r⌋\left\lfloor 10^{5} r\right\rfloor. Note: SS has natural density rr if 1n|S∩{1,…,n}|\frac{1}{n}|S \cap\{1, \ldots, n\}| approaches rr as nn approaches ∞\infty.
Plain-text mathematical notation (without MathML)
Let S be a set of positive integers satisfying the following two conditions: - For each positive integer n, at least one of n,2n,…,100n is in S. - If a₁,a₂,b₁,b₂ are positive integers such that gcd(a₁a₂,b₁b₂)=1 and a₁b₁,a₂b₂∈S, then a₂b₁,a₁b₂∈S Suppose that S has natural density r. Compute the minimum possible value of ⌊10⁵r⌋. Note: S has natural density r if (1)/(n)|S∩{1,…,n}| approaches r as n approaches ∞.
Original LaTeX notation
Let $S$ be a set of positive integers satisfying the following two conditions: - For each positive integer $n$, at least one of $n, 2 n, \ldots, 100 n$ is in $S$. - If $a_{1}, a_{2}, b_{1}, b_{2}$ are positive integers such that $\operatorname{gcd}\left(a_{1} a_{2}, b_{1} b_{2}\right)=1$ and $a_{1} b_{1}, a_{2} b_{2} \in S$, then $a_{2} b_{1}, a_{1} b_{2} \in S$ Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\left\lfloor 10^{5} r\right\rfloor$. Note: $S$ has natural density $r$ if $\frac{1}{n}|S \cap\{1, \ldots, n\}|$ approaches $r$ as $n$ approaches $\infty$.

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Source and history

task_id
886965c6-3232-5ef0-a9e0-195c7d17875b
task_key
test--886965c6-3232-5ef0-a9e0-195c7d17875b
task_revision_id
1
config / split
/ test

Official source

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