# User:Eznutella888

Hello fellow users of AOPS, my name is $\mathbb{EZNUTELLA} \text{888}$! As you can see I like math. That's why I'm here.

I have taken many math competitions, including the Canadian Gauss, Pascal and Cayley. I have also taken Canadian Intermediate Mathematics Examination, and the Math Challengers competition sponsored by the Canadian Math Challengers Society. I also have taken AMC 8, and this year I'm taking the AMC 10, as well as the COMC (Canadian Open Mathematics Challenge).

$[asy] size(6cm); pair D=(0,0), C=(6,0), B=(6,3), A=(0,3); draw(A--B--C--D--cycle); draw(B--D); draw(A--(6,2)); draw(A--(6,1)); label("A", A, dir(135)); label("B", B, dir(45)); label("C", C, dir(-45)); label("D", D, dir(-135)); label("Q", extension(A,(6,1),B,D),dir(-90)); label("P", extension(A,(6,2),B,D), dir(90)); label("F", (6,1), dir(0)); label("E", (6,2), dir(0)); [/asy]$

We can set coordinates for the points. $D=(0,0), C=(6,0), B=(6,3), A=(0,3)$. The line $BD$'s equation is $y = \frac{1}{2}$, $AE$'s equation is $y = -\frac{1}{6}x + 3$, and line $AF$'s equation is $y = -\frac{1}{3}x + 3$. Adding the equations of lines $BD$ and $AE$, the coordinates of $P$ is $(\frac{9}{2},\frac{9}{4})$. Furthermore the coordinates $Q$ is $(\frac{18}{5}, \frac{9}{5})$. Using the Pythagorean Theorem, the length of $QD$ is $\sqrt{(\frac{18}{5})^2+(\frac{9}{5})^2} = \sqrt{\frac{405}{25}} = \frac{\sqrt{405}}{5} = \frac{9\sqrt{5}}{5}$, and the length of $DP$ = $\sqrt{(\frac{9}{2})^2+(\frac{9}{4})^2} = \sqrt{\frac{81}{4} + \frac{81}{16}} = \sqrt{\frac{405}{16}} = \frac{\sqrt{405}}{4} = \frac{9\sqrt{5}}{4}. PQ = DP - DQ = \frac{9\sqrt{5}}{5} - \frac{9\sqrt{5}}{4} = \frac{9\sqrt{5}}{20}. The length of$DB = \sqrt{6^2 + 3^2} = \sqrt{45} = 3\sqrt{5}$. Then$BP= 3\sqrt{5} - \frac{9\sqrt{5}}{4} = frac{3\sqrt{5}}{4}. Then the ratio $BP : PQ : QD: = frac{3\sqrt{5}}{4} : \frac{9\sqrt{5}}{20} : \frac{9\sqrt{5}}{5} = 15\sqrt{5} : 9\sqrt{5} : 36\sqrt{5} = 15 : 9 : 36 = 5 : 3 : 12. Then$r, s,$and$t$is$5, 3,$and$12$, respectively. The problem tells us to find$r + s + t$, so$5 + 3 + 12 = \boxed{\textbf{(E) }20}\$