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Omni-MATH / Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She…

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problem

Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle A0A1A2A_{0} A_{1} A_{2} where angle A0A_{0} is 90∘90^{\circ}, angle A1A_{1} is 60∘60^{\circ}, and A0A1A_{0} A_{1} is 1. She then extends the pasture. First, she extends A2A0A_{2} A_{0} to A3A_{3} such that A3A0=12A2A0A_{3} A_{0}=\frac{1}{2} A_{2} A_{0} and the new pasture is triangle A1A2A3A_{1} A_{2} A_{3}. Next, she extends A3A1A_{3} A_{1} to A4A_{4} such that A4A1=16A3A1A_{4} A_{1}=\frac{1}{6} A_{3} A_{1}. She continues, each time extending AnAn−2A_{n} A_{n-2} to An+1A_{n+1} such that An+1An−2=12n−2AnAn−2A_{n+1} A_{n-2}=\frac{1}{2^{n}-2} A_{n} A_{n-2}. What is the smallest KK such that her pasture never exceeds an area of KK?
Plain-text mathematical notation (without MathML)
Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle A₀A₁A₂ where angle A₀ is 90^(∘), angle A₁ is 60^(∘), and A₀A₁ is 1. She then extends the pasture. First, she extends A₂A₀ to A₃ such that A₃A₀=(1)/(2)A₂A₀ and the new pasture is triangle A₁A₂A₃. Next, she extends A₃A₁ to A₄ such that A₄A₁=(1)/(6)A₃A₁. She continues, each time extending A_(n)A_(n−2) to A_(n+1) such that A_(n+1)A_(n−2)=(1)/(2^(n)−2)A_(n)A_(n−2). What is the smallest K such that her pasture never exceeds an area of K?
Original LaTeX notation
Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle $A_{0} A_{1} A_{2}$ where angle $A_{0}$ is $90^{\circ}$, angle $A_{1}$ is $60^{\circ}$, and $A_{0} A_{1}$ is 1. She then extends the pasture. First, she extends $A_{2} A_{0}$ to $A_{3}$ such that $A_{3} A_{0}=\frac{1}{2} A_{2} A_{0}$ and the new pasture is triangle $A_{1} A_{2} A_{3}$. Next, she extends $A_{3} A_{1}$ to $A_{4}$ such that $A_{4} A_{1}=\frac{1}{6} A_{3} A_{1}$. She continues, each time extending $A_{n} A_{n-2}$ to $A_{n+1}$ such that $A_{n+1} A_{n-2}=\frac{1}{2^{n}-2} A_{n} A_{n-2}$. What is the smallest $K$ such that her pasture never exceeds an area of $K$?

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Source and history

task_id
3c588d29-5caf-55f1-a298-45ecda0b5aea
task_key
test--3c588d29-5caf-55f1-a298-45ecda0b5aea
task_revision_id
1
config / split
/ test

Official source

initial import