MathDB

2022 PUMaC Team #2

Source:

September 9, 2023

geometry

Problem Statement

A triangle

\vartriangle A_0A_1A_2

in the plane has sidelengths

A_0A_1 = 7

,

A_1A_2 = 8

,

A_2A_0 = 9

. For

i \ge 0

, given

\vartriangle A_iA_{i+1}A_{i+2}

, let

A_{i+3}

be the midpoint of

A_iA_{i+1}

and let Gi be the centroid of

\vartriangle A_iA_{i+1}A_{i+2}

. Let point

G

be the limit of the sequence of points

\{G_i\}^{\infty}_{i=0}

. If the distance between

G

and

G_0

can be written as

\frac{a\sqrt{b}}{c}

, where

a, b, c

are positive integers such that

a

and

c

are relatively prime and

b

is not divisible by the square of any prime, find

a^2 + b^2 + c^2

.

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