MathDB
2013 HMMT Guts #10: Wesyu and her Cao Pasture

Source:

March 26, 2013
HMMTgeometry

Problem Statement

Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. Shw starts with a triangle A0A1A2A_0A_1A_2 where angle A0A_0 is 90∘90^\circ, angle A1A_1 is 60∘60^\circ, and A0A1A_0A_1 is 11. She then extends the pasture. FIrst, she extends A2A0A_2A_0 to A3A_3 such that A3A0=12A2A0A_3A_0=\dfrac12A_2A_0 and the new pasture is triangle A1A2A3A_1A_2A_3. Next, she extends A3A1A_3A_1 to A4A_4 such that A4A1=16A3A1A_4A_1=\dfrac16A_3A_1. She continues, each time extending AnAn−2A_nA_{n-2} to An+1A_{n+1} such that An+1An−2=12n−2AnAn−2A_{n+1}A_{n-2}=\dfrac1{2^n-2}A_nA_{n-2}. What is the smallest KK such that her pasture never exceeds an area of KK?
Back to ProblemsView on AoPS