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2021 LMT Spring Division A Problem 20

Source:

October 22, 2021

Problem Statement

Let

\Omega

be a circle with center

O

. Let

\omega_1

and

\omega_2

be circles with centers

O_1

and

O_2

, respectively, internally tangent to

\Omega

at points

A

and

B

, respectively, such that

O_1

is on

\overline{OA}

, and

O_2

is on

\overline{OB}

and

\omega_1

. There exists a point

P

on line

AB

such that

P

is on both

\omega_1

and

\omega_2

. Let the external tangent of

\omega_1

and

\omega_2

on the same side of line

AB

as

O

hit

\omega_1

at

X

and

\omega_2

at

Y

, and let lines

AX

and

BY

intersect at

N

. Given that

O_1X = 81

and

O_2Y = 18

, the value of

NX \cdot NA

can be written as

a\sqrt{b} + c

, where

a

,

b

, and

c

are positive integers, and

b

is not divisible by the square of a prime. Find

a+b+c

.
Proposed by Kevin Zhao

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