MathDB

In the interior of square

ABCD

with side length

1

, a point

P

is chosen such that the lines

\ell_1, \ell_2

through

P

parallel to

AC

and

BD

, respectively, divide the square into four distinct regions, the smallest of which has area

\mathcal{R}

. The area of the region of all points

P

for which

\mathcal{R} \geq \tfrac{1}{6}

can be expressed as

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

where

\gcd(a,b,d)=1

and

c

is not divisible by the square of any prime. Compute

a+b+c+d

.
Proposed by Andrew Wen

20