MathDB

The vertical cross section of a circular cone with vertex

P

is an isoceles right triangle. Point

A

is on the base circle, point

B

is interior to the base circle,

O

is the center of the base circle,

AB \perp OB

B

OH \perp PB

H

PA = 4

, and

C

is the midpoint of

PA

. When the volume of tetrahedron

OHPC

is maximized, the length of

OB

x

x^2

is in the form

\frac{p}{q}

where

p, q

are relatively prime positive integers. Find

p + q

8