MathDB

Hexagon

ABCDEF

is divided into four rhombuses,

\mathcal{P, Q, R, S,}

and

\mathcal{T,}

as shown. Rhombuses

\mathcal{P, Q, R,}

and

\mathcal{S}

are congruent, and each has area

\sqrt{2006}

. Let

K

be the area of rhombus

\mathcal{T}

. Given that

K

is a positive integer, find the number of possible values for

K

.
[asy] size(150);defaultpen(linewidth(0.7)+fontsize(10)); draw(rotate(45)*polygon(4)); pair F=(1+sqrt(2))*dir(180), C=(1+sqrt(2))*dir(0), A=F+sqrt(2)*dir(45), E=F+sqrt(2)*dir(-45), B=C+sqrt(2)*dir(180-45), D=C+sqrt(2)*dir(45-180); draw(F--(-1,0)^^C--(1,0)^^A--B--C--D--E--F--cycle); pair point=origin; label("

A

", A, dir(point--A)); label("

B

", B, dir(point--B)); label("

C

", C, dir(point--C)); label("

D

", D, dir(point--D)); label("

E

", E, dir(point--E)); label("

F

", F, dir(point--F)); label("

\mathcal{P}

", intersectionpoint( A--(-1,0), F--(0,1) )); label("

\mathcal{S}

", intersectionpoint( E--(-1,0), F--(0,-1) )); label("

\mathcal{R}

", intersectionpoint( D--(1,0), C--(0,-1) )); label("

\mathcal{Q}

", intersectionpoint( B--(1,0), C--(0,1) )); label("

\mathcal{T}

", point); dot(A^^B^^C^^D^^E^^F);[/asy]

8

Problems(2)