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Omni-MATH / Given a acute triangle PA₁B₁ is inscribed in the circle Γ with radius 1. for all integers n≥1 are defined: C_(n) the foot of the perpendicular from P to A_(n)B_(n) O_(n) is the cen…

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Given a acute triangle PA1B1PA_1B_1 is inscribed in the circle Γ\Gamma with radius 11. for all integers n≥1n \ge 1 are defined: CnC_n the foot of the perpendicular from PP to AnBnA_nB_n OnO_n is the center of ⊙(PAnBn)\odot (PA_nB_n) An+1A_{n+1} is the foot of the perpendicular from CnC_n to PAnPA_n Bn+1≡PBn∩OnAn+1B_{n+1} \equiv PB_n \cap O_nA_{n+1} If PC1=2PC_1 =\sqrt{2}, find the length of PO2015PO_{2015} [hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]
Plain-text mathematical notation (without MathML)
Given a acute triangle PA₁B₁ is inscribed in the circle Γ with radius 1. for all  integers n≥1 are defined:
C_(n) the foot of the perpendicular from P to A_(n)B_(n)
O_(n) is the center of ⊙(PA_(n)B_(n))
A_(n+1) is the foot of the perpendicular from C_(n) to PA_(n)
B_(n+1)≡PB_(n)∩O_(n)A_(n+1)

If PC₁=√(2), find the length of PO₂₀₁₅ 

[hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]
Original LaTeX notation
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all  integers $n \ge 1$ are defined:
$C_n$ the foot of the perpendicular from $P$ to $A_nB_n$
$O_n$ is the center of $\odot (PA_nB_n)$
$A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$
$B_{n+1} \equiv PB_n \cap O_nA_{n+1}$

If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$ 

[hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]

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task_id
2b927459-983d-5498-bd30-45fe68e60a6c
task_key
test--2b927459-983d-5498-bd30-45fe68e60a6c
task_revision_id
1
config / split
/ test

Official source

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