MathDB

2016-2017 Fall OMO Problem 13

Source:

November 16, 2016

Online Math Open

Problem Statement

Let

A_1B_1C_1

be a triangle with

A_1B_1 = 16, B_1C_1 = 14,

and

C_1A_1 = 10

. Given a positive integer

i

and a triangle

A_iB_iC_i

with circumcenter

O_i

, define triangle

A_{i+1}B_{i+1}C_{i+1}

in the following way:
(a)

A_{i+1}

is on side

B_iC_i

such that

C_iA_{i+1}=2B_iA_{i+1}

. (b)

B_{i+1}\neq C_i

is the intersection of line

A_iC_i

with the circumcircle of

O_iA_{i+1}C_i

. (c)

C_{i+1}\neq B_i

is the intersection of line

A_iB_i

with the circumcircle of

O_iA_{i+1}B_i

.
Find

\left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2.

Note:

[K]

denotes the area of

K

.
Proposed by Yang Liu

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