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Omni-MATH / In a fictional world, each resident (viewed as geometric point) is assigned a…

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problem

In a fictional world, each resident (viewed as geometric point) is assigned a number: 1,2,⋯1,2, \cdots. In order to fight against some epidemic, the residents take some vaccine and they stay at the vaccination site after taking the shot for observation. Now suppose that the shape of the Observation Room is a circle of radius 14\frac{1}{4}, and one requires that the distance dm,nd_{m, n} between the Resident No. mm and the Resident No. nn must satisfy (m+n)dm,n≥1(m+n) d_{m, n} \geq 1. Where we consider the distance on the circle, i.e., the length of the minor arc between two points. Proof Question: Give a proof of your answer to Question (i).
Plain-text mathematical notation (without MathML)
In a fictional world, each resident (viewed as geometric point) is assigned a number: 1,2,⋯. In order to fight against some epidemic, the residents take some vaccine and they stay at the vaccination site after taking the shot for observation. Now suppose that the shape of the Observation Room is a circle of radius (1)/(4), and one requires that the distance d_(m,n) between the Resident No. m and the Resident No. n must satisfy (m+n)d_(m,n)≥1. Where we consider the distance on the circle, i.e., the length of the minor arc between two points. Proof Question: Give a proof of your answer to Question (i).
Original LaTeX notation
In a fictional world, each resident (viewed as geometric point) is assigned a number: $1,2, \cdots$. In order to fight against some epidemic, the residents take some vaccine and they stay at the vaccination site after taking the shot for observation. Now suppose that the shape of the Observation Room is a circle of radius $\frac{1}{4}$, and one requires that the distance $d_{m, n}$ between the Resident No. $m$ and the Resident No. $n$ must satisfy $(m+n) d_{m, n} \geq 1$. Where we consider the distance on the circle, i.e., the length of the minor arc between two points. Proof Question: Give a proof of your answer to Question (i).

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Source and history

task_id
4d1d1673-67e1-59e0-92fb-f69521d3f609
task_key
test--4d1d1673-67e1-59e0-92fb-f69521d3f609
task_revision_id
1
config / split
/ test

Official source

initial import