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Omni-MATH / One day, there is a Street Art Show at somewhere, and there are some spectators…
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One day, there is a Street Art Show at somewhere, and there are some spectators around. We consider this place as an Euclidean plane. Let be the center of the show. And name the spectators by They pick their positions one by one. The positions need to satisfy the following three conditions simultaneously. (i) The distance between and is no less than 10 meters, that is,
$K P_{n} \geq 10 \mathrm{~m}$ holds for any positive integer . (ii) The distance between and any previous spectator is no less than 1 meter, that is, $P_{m} P_{n} \geq 1 \mathrm{~m}$ holds for any and any . (iii) always choose the position closest to that satisfies (i) and (ii), that is, reaches its minimum possible value. If there are more than one point that satisfy (i) and (ii) and have the minimum distance to may choose any one of them. For example, is not restricted by (ii), so he may choose any point on the circle which is centered at with radius 10 meters. For , since there are lots of points on which are at least 1 meter apart from , he may choose anyone of them. (1) Which of the following statement is true? (A) There exist positive real numbers such that for any positive integer , no matter how choose their positions, always hold (unit: meter); (B) There exist positive real numbers such that for any positive integer , no matter how choose their positions, always hold (unit: meter); (C) There exist positive real numbers such that for any positive integer , no matter how choose their positions, always hold (unit: meter); (D) There exist positive real numbers such that for any positive integer , no matter how choose their positions, always hold (unit: meter).Plain-text mathematical notation (without MathML)
One day, there is a Street Art Show at somewhere, and there are some spectators around. We consider this place as an Euclidean plane. Let K be the center of the show. And name the spectators by A₁,A₂,…,A_(n),… They pick their positions P₁,P₂,…,P_(n),… one by one. The positions need to satisfy the following three conditions simultaneously. (i) The distance between K and A_(n) is no less than 10 meters, that is, $K P_{n} \geq 10 \mathrm{~m}$ holds for any positive integer n. (ii) The distance between A_(n) and any previous spectator is no less than 1 meter, that is, $P_{m} P_{n} \geq 1 \mathrm{~m}$ holds for any n≥2 and any 1≤m≤n−1. (iii) A_(n) always choose the position closest to K that satisfies (i) and (ii), that is, KP_(n) reaches its minimum possible value. If there are more than one point that satisfy (i) and (ii) and have the minimum distance to K,A_(n) may choose any one of them. For example, A₁ is not restricted by (ii), so he may choose any point on the circle C which is centered at K with radius 10 meters. For A₂, since there are lots of points on C which are at least 1 meter apart from P₁, he may choose anyone of them. (1) Which of the following statement is true? (A) There exist positive real numbers c₁,c₂ such that for any positive integer n, no matter how A₁,A₂,…,A_(n) choose their positions, c₁≤KP_(n)≤c₂ always hold (unit: meter); (B) There exist positive real numbers c₁,c₂ such that for any positive integer n, no matter how A₁,A₂,…,A_(n) choose their positions, c₁√(n)≤KP_(n)≤c₂√(n) always hold (unit: meter); (C) There exist positive real numbers c₁,c₂ such that for any positive integer n, no matter how A₁,A₂,…,A_(n) choose their positions, c₁n≤KP_(n)≤c₂n always hold (unit: meter); (D) There exist positive real numbers c₁,c₂ such that for any positive integer n, no matter how A₁,A₂,…,A_(n) choose their positions, c₁n²≤KP_(n)≤c₂n² always hold (unit: meter).Original LaTeX notation
One day, there is a Street Art Show at somewhere, and there are some spectators around. We consider this place as an Euclidean plane. Let $K$ be the center of the show. And name the spectators by $A_{1}, A_{2}, \ldots, A_{n}, \ldots$ They pick their positions $P_{1}, P_{2}, \ldots, P_{n}, \ldots$ one by one. The positions need to satisfy the following three conditions simultaneously. (i) The distance between $K$ and $A_{n}$ is no less than 10 meters, that is, $K P_{n} \geq 10 \mathrm{~m}$ holds for any positive integer $n$. (ii) The distance between $A_{n}$ and any previous spectator is no less than 1 meter, that is, $P_{m} P_{n} \geq 1 \mathrm{~m}$ holds for any $n \geq 2$ and any $1 \leq m \leq n-1$. (iii) $A_{n}$ always choose the position closest to $K$ that satisfies (i) and (ii), that is, $K P_{n}$ reaches its minimum possible value. If there are more than one point that satisfy (i) and (ii) and have the minimum distance to $K, A_{n}$ may choose any one of them. For example, $A_{1}$ is not restricted by (ii), so he may choose any point on the circle $C$ which is centered at $K$ with radius 10 meters. For $A_{2}$, since there are lots of points on $C$ which are at least 1 meter apart from $P_{1}$, he may choose anyone of them. (1) Which of the following statement is true? (A) There exist positive real numbers $c_{1}, c_{2}$ such that for any positive integer $n$, no matter how $A_{1}, A_{2}, \ldots, A_{n}$ choose their positions, $c_{1} \leq K P_{n} \leq c_{2}$ always hold (unit: meter); (B) There exist positive real numbers $c_{1}, c_{2}$ such that for any positive integer $n$, no matter how $A_{1}, A_{2}, \ldots, A_{n}$ choose their positions, $c_{1} \sqrt{n} \leq K P_{n} \leq c_{2} \sqrt{n}$ always hold (unit: meter); (C) There exist positive real numbers $c_{1}, c_{2}$ such that for any positive integer $n$, no matter how $A_{1}, A_{2}, \ldots, A_{n}$ choose their positions, $c_{1} n \leq K P_{n} \leq c_{2} n$ always hold (unit: meter); (D) There exist positive real numbers $c_{1}, c_{2}$ such that for any positive integer $n$, no matter how $A_{1}, A_{2}, \ldots, A_{n}$ choose their positions, $c_{1} n^{2} \leq K P_{n} \leq c_{2} n^{2}$ always hold (unit: meter).Discussion
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