MathDB

Let

ABC

be an acute triangle with

\cos B =\frac{1}{3}, \cos C =\frac{1}{4}

, and circumradius

72

. Let

ABC

have circumcenter

O

, symmedian point

K

, and nine-point center

N

. Consider all non-degenerate hyperbolas

\mathcal H

with perpendicular asymptotes passing through

A,B,C

. Of these

\mathcal H

, exactly one has the property that there exists a point

P\in \mathcal H

such that

NP

is tangent to

\mathcal H

and

P\in OK

. Let

N'

be the reflection of

N

over

BC

. If

AK

meets

PN'

Q

, then the length of

PQ

can be expressed in the form

a+b\sqrt{c}

, where

a,b,c

are positive integers such that

c

is not divisible by the square of any prime. Compute

100a+b+c

.
Proposed by Vincent Huang

Problem Statement