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Maximize [BKL] where KL passes through a fixed point A

Source: AIME II 2013, Problem 10

April 4, 2013

geometryLaTeXnumber theoryrelatively primeAMCAIME

Problem Statement

Given a circle of radius

\sqrt{13}

, let

A

be a point at a distance

4 + \sqrt{13}

from the center

O

of the circle. Let

B

be the point on the circle nearest to point

A

. A line passing through the point

A

intersects the circle at points

K

and

L

. The maximum possible area for

\triangle BKL

can be written in the form

\tfrac{a-b\sqrt{c}}{d}

, where

a

,

b

,

c

, and

d

are positive integers,

a

and

d

are relatively prime, and

c

is not divisible by the square of any prime. Find

a+b+c+d

.

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