AIME 2015 II · 第 15 题

Solution 1 (guys trig is fast)

Let $M$ be the intersection of $\overline{BC}$ and the common internal tangent of $\mathcal P$ and $\mathcal Q.$ We claim that $M$ is the circumcenter of right $\triangle{ABC}.$ Indeed, we have $AM = BM$ and $BM = CM$ by equal tangents to circles, and since $BM = CM, M$ is the midpoint of $\overline{BC},$ implying that $\angle{BAC} = 90.$ Now draw $\overline{PA}, \overline{PB}, \overline{PM},$ where $P$ is the center of circle $\mathcal P.$ Quadrilateral $PAMB$ is cyclic, and by Pythagorean Theorem $PM = \sqrt{5},$ so by Ptolemy on $PAMB$ we have

$$AB \sqrt{5} = 2 \cdot 1 + 2 \cdot 1 = 4 \iff AB = \dfrac{4 \sqrt{5}}{5}.$$ Do the same thing on cyclic quadrilateral $QAMC$ (where $Q$ is the center of circle $\mathcal Q$ and get $AC = \frac{8 \sqrt{5}}{5}.$

Let $\angle A = \angle{DAB}.$ By Law of Sines, $BD = 2R \sin A = 2 \sin A.$ Note that $\angle{D} = \angle{ABC}$ from inscribed angles, so

$$\begin{aligned} [ABD] &= \dfrac{1}{2} BD \cdot AB \cdot \sin{\angle B} \\ &= \dfrac{1}{2} \cdot \dfrac{4 \sqrt{5}}{5} \cdot 2 \sin A \sin{\left(180 - \angle A - \angle D\right)} \\ &= \dfrac{4 \sqrt{5}}{5} \cdot \sin A \cdot \sin{\left(\angle A + \angle D\right)} \\ &= \dfrac{4 \sqrt{5}}{5} \cdot \sin A \cdot \left(\sin A \cos D + \cos A \sin D\right) \\ &= \dfrac{4 \sqrt{5}}{5} \cdot \sin A \cdot \left(\sin A \cos{\angle{ABC}} + \cos A \sin{\angle{ABC}}\right) \\ &= \dfrac{4 \sqrt{5}}{5} \cdot \sin A \cdot \left(\dfrac{\sqrt{5} \sin A}{5} + \dfrac{2 \sqrt{5} \cos A}{5}\right) \\ &= \dfrac{4}{5} \cdot \sin A \left(\sin A + 2 \cos A\right) \end{aligned}$$ after angle addition identity.

Similarly, $\angle{EAC} = 90 - \angle A,$ and by Law of Sines $CE = 8 \sin{\angle{EAC}} = 8 \cos A.$ Note that $\angle{E} = \angle{ACB}$ from inscribed angles, so

$$\begin{aligned} [ACE] &= \dfrac{1}{2} AC \cdot CE \sin{\angle C} \\ &= \dfrac{1}{2} \cdot \dfrac{8 \sqrt{5}}{5} \cdot 8 \cos A \sin{\left[180 - \left(90 - \angle A\right) - \angle E\right]} \\ &= \dfrac{32 \sqrt{5}}{5} \cdot \cos A \sin{\left[\left(90 - \angle A\right) + \angle{ACB}\right]} \\ &= \dfrac{32 \sqrt{5}}{5} \cdot \cos A \left(\dfrac{2 \sqrt{5} \cos A}{5} + \dfrac{\sqrt{5} \sin A}{5}\right) \\ &= \dfrac{32}{5} \cdot \cos A \left(\sin A + 2 \cos A\right) \end{aligned}$$ after angle addition identity. Setting the two areas equal, we get

$$\tan A = \frac{\sin A}{\cos A} = 8 \iff \sin A = \frac{8}{\sqrt{65}}, \cos A = \frac{1}{\sqrt{65}}$$ after Pythagorean Identity. Now plug back in and the common area is $\frac{64}{65} \iff \boxed{129}.$

Solution 2

Call $O_1$ and $O_2$ the centers of circles $\mathcal{P}$ and $\mathcal{Q}$, respectively, and extend $CB$ and $O_2O_1$ to meet at point $N$. Call $K$ and $L$ the feet of the altitudes from $B$ to $O_1N$ and $C$ to $O_2N$, respectively. Using the fact that $\triangle{O_1BN} \sim \triangle{O_2CN}$ and setting $NO_1 = k$, we have that $\frac{k+5}{k} = \frac{4}{1} \implies k=\frac{5}{3}$. We can do some more length chasing using triangles similar to $O_1BN$ to get that $AK = AL = \frac{24}{15}$, $BK = \frac{12}{15}$, and $CL = \frac{48}{15}$. Now, consider the circles $\mathcal{P}$ and $\mathcal{Q}$ on the coordinate plane, where $A$ is the origin. If the line $\ell$ through $A$ intersects $\mathcal{P}$ at $D$ and $\mathcal{Q}$ at $E$ then $4 \cdot DA = AE$. To verify this, notice that $\triangle{AO_1D} \sim \triangle{EO_2A}$ from the fact that both triangles are isosceles with $\angle{O_1AD} \cong \angle{O_2AE}$, which are corresponding angles. Since $O_2A = 4\cdot O_1A$, we can conclude that $4 \cdot DA = AE$.

Hence, we need to find the slope $m$ of line $\ell$ such that the perpendicular distance $n$ from $B$ to $AD$ is four times the perpendicular distance $p$ from $C$ to $AE$. This will mean that the product of the bases and heights of triangles $ACE$ and $DBA$ will be equal, which in turn means that their areas will be equal. Let the line $\ell$ have the equation $y = -mx \implies mx + y = 0$, and let $m$ be a positive real number so that the negative slope of $\ell$ is preserved. Setting $A = (0,0)$, the coordinates of $B$ are $(x_B, y_B) = \left(\frac{-24}{15}, \frac{-12}{15}\right)$, and the coordinates of $C$ are $(x_C, y_C) = \left(\frac{24}{15}, \frac{-48}{15}\right)$. Using the point-to-line distance formula and the condition $n = 4p$, we have

$$\frac{|mx_B + 1(y_B) + 0|}{\sqrt{m^2 + 1}} = \frac{4|mx_C + 1(y_C) + 0|}{\sqrt{m^2 + 1}}$$ $$\implies |mx_B + y_B| = 4|mx_C + y_C| \implies \left|\frac{-24m}{15} + \frac{-12}{15}\right| = 4\left|\frac{24m}{15} + \frac{-48}{15}\right|.$$ If $m > 2$, then clearly $B$ and $C$ would not lie on the same side of $\ell$. Thus since $m > 0$, we must switch the signs of all terms in this equation when we get rid of the absolute value signs. We then have

$$\frac{6m}{15} + \frac{3}{15} = \frac{48}{15} - \frac{24m}{15} \implies 2m = 3 \implies m = \frac{3}{2}.$$ Thus, the equation of $\ell$ is $y = -\frac{3}{2}x$.

Then we can find the coordinates of $D$ by finding the point $(x,y)$ other than $A = (0,0)$ where the circle $\mathcal{P}$ intersects $\ell$. $\mathcal{P}$ can be represented with the equation $(x + 1)^2 + y^2 = 1$, and substituting $y = -\frac{3}{2}x$ into this equation yields $x = 0, -\frac{8}{13}$ as solutions. Discarding $x = 0$, the $y$-coordinate of $D$ is $-\frac{3}{2} \cdot -\frac{8}{13} = \frac{12}{13}$. The distance from $D$ to $A$ is then $\frac{4}{\sqrt{13}}.$ The perpendicular distance from $B$ to $AD$ or the height of $\triangle{DBA}$ is $\frac{|\frac{3}{2}\cdot\frac{-24}{15} + \frac{-12}{15} + 0|}{\sqrt{\frac{3}{2}^2 + 1}} = \frac{\frac{48}{15}}{\frac{\sqrt{13}}{2}} = \frac{32}{5\sqrt{13}}.$ Finally, the common area is $\frac{1}{2}\left(\frac{32}{5\sqrt{13}} \cdot \frac{4}{\sqrt{13}}\right) = \frac{64}{65}$, and $m + n = 64 + 65 = \boxed{129}$.

Solution 3

By homothety, we deduce that $AE = 4 AD$. (The proof can also be executed by similar triangles formed from dropping perpendiculars from the centers of $P$ and $Q$ to $l$.) Therefore, our equality of area condition, or the equality of base times height condition, reduces to the fact that the distance from $B$ to $l$ is four times that from $C$ to $l$. Let the distance from $C$ be $x$ and the distance from $B$ be $4x$.

Let $P$ and $Q$ be the centers of their respective circles. Then dropping a perpendicular from $P$ to $Q$ creates a $3-4-5$ right triangle, from which $BC = 4$ and, if $\alpha = \angle{AQC}$, that $\cos \alpha = \dfrac{3}{5}$. Then $\angle{BPA} = 180^\circ - \alpha$, and the Law of Cosines on triangles $APB$ and $AQC$ gives $AB = \dfrac{4}{\sqrt{5}}$ and $AC = \dfrac{8}{\sqrt{5}}.$

Now, using the Pythagorean Theorem to express the length of the projection of $BC$ onto line $l$ gives

$$\sqrt{\frac{16}{5} - 16x^2} + \sqrt{\frac{64}{5} - x^2} = \sqrt{16 - 9x^2}.$$ Squaring and simplifying gives

$$\sqrt{\left(\frac{1}{5} - x^2\right)\left(\frac{64}{5} - x^2\right)} = x^2,$$ and squaring and solving gives $x = \dfrac{8}{5\sqrt{13}}.$

By the Law of Sines on triangle $ABD$, we have

$$\frac{BD}{\sin A} = 2.$$ But we know $\sin A = \dfrac{4x}{AB}$, and so a small computation gives $BD = \dfrac{16}{\sqrt{65}}.$ The Pythagorean Theorem now gives

$$AD = \sqrt{BD^2 - (4x)^2} + \sqrt{AB^2 - (4x)^2} = \frac{4}{\sqrt{13}},$$ and so the common area is $\dfrac{1}{2} \cdot \frac{4}{\sqrt{13}} \cdot \frac{32}{5\sqrt{13}} = \frac{64}{65}.$ The answer is $\boxed{129}.$

Alternate Path to x

Call the intersection of lines $l$ and $BC$ $E$.You can use similar triangles to find that the distance from $B$ to $E$ is four times the distance from $C$ to $E$. Then draw a perpendicular from $A$ to $BC$ and call the point $F$. $AF = \frac{8}{5}$ and $FE = FC + CE = \frac{16}{5} + \frac{4}{3} = \frac{68}{15}$, so by the Pythagorean Theorem, $AE = \dfrac{20\sqrt{13}}{5}$. You can now use similar triangles to find that $x = \dfrac{8}{5\sqrt{13}}$ and continue on like in solution 2.

Solution 4

$DE$ goes through $A$, the point of tangency of both circles. So $DE$ intercepts equal arcs in circle $P$ and $Q$: homothety. Hence, $AE=4AD$. We will use such similarity later.

The diagonal distance between the centers of the circles is $4+1=5$. The difference in heights is $4-1=3$. So $BC=\sqrt{5^2-3^2}=4$.

The triangle connecting the centers with a side parallel to $BC$ is a $3-4-5$ right triangle. Since $O_PA=1$, the height of $A$ is $1+3/5=8/5$. Drop an altitude from $A$ to $BC$ and call it $I$: $IB=4/5$ and $IC=4-4/5=16/5$. Since right $\triangle AIB\sim\triangle CIB$, $ABC$ is a right triangle also; $IB:IA:IC$ form a geometric progression $\times 2$.

Extend $BA$ through $A$ to a point $G$ on the other side of $\circ Q$. By homothety, $\triangle DAB\sim\triangle EAG$. By angle chasing $\triangle DAB$ through right triangle $ABC$, we deduce that $\angle CEG$ is a right angle. Since $ACEG$ is cyclic, $\angle GAC$ is also right. So $CG$ is a diameter of $\circ G$. Because of this, $CG \perp BC$, the tangent line. $\triangle BCG$ is right and $\triangle BCG\sim\triangle ABC\sim\triangle CAG$.

$AC=\sqrt{(8/5)^2+(16/5)^2}=8\sqrt{5}/5$ so $AG=2AC=16\sqrt{5}/5$ and $[\triangle CAG]=64/5$.

Since $[\triangle DAB]=[\triangle ACE]$, the common area is $[ACEG]/17$. $16[\triangle DAB]=[\triangle GAE]$ because the triangles are similar with a ratio of $1:4$. So we only need to find $[\triangle CEG]$ now.

Extend $DE$ through $E$ to intersect the tangent at $F$. Because $4DA=AE$, the altitude from $B$ to $AD$ is $4$ times the height from $C$ to $EA$. So $BC=3/4BF$ and $BF=16/3$. We look at right triangle $\triangle AIF$. $IF=68/15$ and $AI=8/5$. $\triangle AIF$ is a $17-6-5\sqrt{13}$ right triangle. Hypotenuse $AF$ intersects $CG$ at a point, we call it $H$. $CH=4/3\div 68/15\cdot 8/5=8/17$. So $HG=8-8/17=128/17$.

By Power of a Point, $CH\cdot HG=AH\cdot HE$. $AH=16/5\cdot 5\sqrt{13}/17=16\sqrt{13}/17.$ So $HE=1024/289\cdot 17/(16\sqrt{13})=64/(17\sqrt{13})$. The height from $E$ to $CG$ is $17/(5\sqrt{13})\cdot 64/(17\sqrt{13})=64/65$.

Thus, $[\triangle CEG]=64/65\cdot 8\div 2=256/65$. The area of the whole cyclic quadrilateral is $64/5+256/65=(832+256)/65=1088/65$. Lastly, the common area is $1/17$ the area of the quadrilateral, or $64/65$. So $64+65=\boxed{129}$.

Solution 5 (HARD computation)

Consider the homothety that takes triangle BDA onto CXY on the big circle, as plotted. Some hidden congruence angles are revealed which help reduce computation complexity. Just some angle chasing and straight forward trigs. Because $AE=XY$ and $AE \parallel XY$, $XYE$ is right angle.

First, $\frac{NO_1}{NO_1+5} = \frac{1}{4}$, so $NO_1=\frac{5}{3}$. And,

$$\cos{\angle{AO_2C}}=\cos{2\angle{AYC}} = \frac{O_2C}{NO_1+5} = \frac{3}{5}$$ $$\sin{\angle{AYC}} = \sqrt{\dfrac{1-\cos{2\angle{AYC}}}{2}}=\frac{1}{\sqrt{5}}$$ $$\cos{\angle{AYC}} = \frac{2}{\sqrt{5}}$$ Then,

$$[AEC] = \frac{1}{2}AE*CE*\sin{\angle{ACE}}=\frac{1}{2}AE*8\sin{\angle{CYE}}*\frac{1}{\sqrt{5}}$$ $$[CXY] = \frac{1}{2}CX*XY*\sin{\angle{CXY}}=\frac{1}{2}*8\sin{\angle{XYC}}*XY*\sin{\angle{CAY}}$$ Since $\angle{CAY} = 90 - \angle{AYC}$, $\angle{XYC} = 90 - \angle{CYE}$, $XY = AE$, we have

$$[CXY] = \frac{1}{2}*8AE\cos{\angle{CYE}}*\cos{\angle{AYC}}=\frac{1}{2}*8AE\cos{\angle{CYE}}*\frac{2}{\sqrt{5}}$$ Since $\triangle{CXY}$ is four times in scale to $\triangle{AEC}$, their area ratio is 16. Divide the two equations for the two areas, we have

$$\tan{\angle{CYE}} = \frac{1}{8}$$ With this angle found, everything else just follows.

$$\sin{\angle{CYE}} = \dfrac{\tan{\angle{CYE}}}{\sqrt{1+\tan^2{\angle{CYE}}}}=\dfrac{1}{\sqrt{65}}$$ $$\cos{\angle{CYE}} = \dfrac{8}{\sqrt{65}}$$ $$\sin{\angle{AYE}} = \sin({\angle{AYC}+\angle{CYE}} )= \frac{1}{\sqrt{5}}*\frac{8}{\sqrt{65}} + \frac{2}{\sqrt{5}}*\frac{1}{\sqrt{65}} = \frac{2}{\sqrt{13}}$$ $$AE = 8\sin{\angle{AYE}} = \frac{16}{\sqrt{13}}$$ $$[AEC] = \frac{1}{2}*8*\frac{16}{\sqrt{13}}*\dfrac{1}{\sqrt{65}}*\frac{1}{\sqrt{5}}=\frac{64}{65}$$ Thus, our answer is $64+65=\boxed{129}$.

Solution 6 (Simple computation)

Let $K$ be the intersection of $BC$ and $AE$. Since the radii of the two circles are 1:4, so we have $AD:AE=1:4$, and the distance from $B$ to line $l$ and the distance from $C$ to line $l$ are in a ratio of 4:1, so $BK:CK=4:1$. We can easily calculate the length of $BC$ to be 4, so $CK=\frac{4}{3}$. Let $J$ be the foot of perpendicular line from $A$ to $BC$, we can know that $BJ:CJ=1:4$, so $BJ = 0.8$, $CJ=3.2$, $AJ=1.6$, and $AK=\sqrt{1.6^2+\left(3.2+\frac{4}{3}\right)^2}=\frac{4}{3}\sqrt{13}$. Since $CK^2 = EK\cdot AK$, so $EK=\frac{4}{39}\sqrt{13}$, and $AE = \frac{4}{3}\sqrt{13} - \frac{4}{39}\sqrt{13}= \frac{16}{13}\sqrt{13}$. $\sin\angle AKB=\frac{AJ}{AK} = \frac{1.6}{\frac{4}{3}\sqrt{13}}=\frac{1.2}{\sqrt{13}}$, so the distance from $C$ to line $l$ is $d=CK\cdot \sin\angle AKB = \frac{4}{3}\cdot \frac{1.2}{\sqrt{13}}=\frac{1.6}{\sqrt{13}}$. so the area is

$$[ACE] = \frac{1}{2}\cdot AE\cdot d = \frac{1}{2}\cdot\frac{16}{13}\sqrt{13}\frac{1.6}{\sqrt{13}} = \frac{64}{65}$$ The final answer is $\boxed{129}$.

--- by Dan Li

Solution 7

Consider the common tangent from $A$ to both circles. Let this intersect $BC$ at point $K$. From equal tangents, we have $BK=AK=CK$, which implies that $\angle BAC = 90^\circ$.

Let the center of $\mathcal{P}$ be $O_1$, and the center of $\mathcal{Q}$ be $O_2$. Angle chasing, we find that $\triangle O_1DA \sim \triangle O_2EA$ with a ratio of $1:4$. Hence $4AD = AE$.

We can easily deduce that $BC=4$ by dropping an altitude from $O_1$ to $O_2C$. Let $\angle ABC = \theta$. By some simple angle chasing, we obtain that $\angle BO_1A = 2\angle BDA = 2\angle ABC = 2\theta,$ and similarly $\angle CO_2A = 180 - 2\theta$.

Using LoC, we get that $AB = \sqrt{2-2\cos2\theta}$ and $AC = \sqrt{32+32\cos2\theta}$. From Pythagorean theorem, we have

$$AB^2 + AC^2 = BC^2 \implies \cos 2\theta = -\frac{3}{5} \implies \cos \theta = \frac{1}{\sqrt5}, \sin \theta = \frac{2}{\sqrt 5}$$ In other words, $AB = \frac{4}{\sqrt5}, AC = \frac{8}{\sqrt5}$.

Using the area condition, we have:

$$\begin{aligned} \frac12 AD*AB \sin \angle DAB &= \frac 12 AE*AC \sin(90-\angle DAB) \\ AD*\frac{4}{\sqrt5} \sin \angle DAB &= 4AD*\frac{8}{\sqrt5} \cos \angle DAB \\ \sin \angle DAB &= 8 \cos \angle DAB \\ \implies \sin \angle DAB &= \frac{8}{\sqrt{65}} \end{aligned}$$ Now, for brevity, let $\angle D = \angle ADB$ and $\angle A = \angle DAB$.

From Law of Sines on $\triangle ABD$, we have

$$\begin{aligned} \frac{AB}{\sin \angle D} &= \frac{AD}{\sin (180-\angle A - \angle D)} \\ \frac{\frac{4}{\sqrt5}}{\frac{2}{\sqrt5}} &= \frac{AD}{\sin \angle A \cos\angle D + \sin \angle D\cos\angle A} \\ 2 &= \frac{AD}{\frac{2}{\sqrt{13}}} \\ AD &= \frac{4}{\sqrt{13}} \end{aligned}$$ It remains to find the area of $\triangle ABD$. This is just

$$\frac12 AD*AB*\sin \angle A = \frac12 * \frac{4}{\sqrt{13}}*\frac{4}{\sqrt5}*\frac{8}{\sqrt{65}} = \frac{64}{65}$$ for an answer of $\boxed{129}.$

This solution was brought to you by Leonard_my_dude.

Solution 8 (Synthetic-Trigonometry)

Add in the line $k$ as the internal tangent between the two circles. Let $M$ be the midpoint of $BC$; It is well-known that $M$ is on $k$ and because $k$ is the radical axis of the two circles, $AM=BM=CM$. Therefore because $M$ is the circumcenter of $\triangle{BAC}$, $\angle{BAC}=90^{\circ}$. Let $O_P$ be the center of circle $\mathcal{P}$ and likewise let $O_Q$ be the center of circle $\mathcal{Q}$. It is well known that by homothety $O_P, A,$ and $O_Q$ are collinear. It is well-known that $\angle{ABC}=\angle{ADB}=b$, and likewise $\angle{ACB}=\angle{AEC}=c$. By homothety, $AD=4AE$, therefore since the two triangles mentioned in the problem, the length of the altitude from $B$ to $AD$ is four times the length of the altitude from $C$ to $AE$. Using the Pythagorean Theorem, $BC = 4$. By angle-chasing, $O_{P}AMB$ is cyclic, and likewise $O_{Q}CMA$ is cyclic. Use the Pythagorean Theorem for $\triangle{O_{P}BM}$ to get $O_{P}M=\sqrt{5}$. Then by Ptolemy's Theorem $AB=\frac{4\sqrt{5}}{5} \implies AC=\frac{8\sqrt{5}}{5}$. Now to compute the area, using what we know about the length of the altitude from $B$ to $AD$ is four times the length of the altitude from $C$ to $AE$, letting $x$ be the length of the altitude from $C$ to $AE$, $x=\frac{8}{5\sqrt{13}}$. From the Law of Sines, $\frac{BD}{\sin{A}}=2 \implies \sin{A}=\frac{4x}{AB} \implies BD=\frac{8x}{AB}=\frac{16\sqrt{65}}{65}$. Then use the Pythagorean Theorem twice and add up the lengths to get $AD=\frac{4}{\sqrt{13}}$. Use the formula $\frac{b \times h}{2}$ to get $\frac{64}{65} = \boxed{129}$ as the answer.

~First

Solution 9 (Visual)

Let $P$ and $Q$ be the centers of circles $\mathcal{P}$ and $\mathcal{Q}$ , respectively.

Let $M$ be midpoint $BC, \beta = \angle ACB.$

Upper diagram shows that

$\sin 2\beta = \frac {4}{5}$ and $AC = 2 AB.$ Therefore $\cos 2\beta = \frac {3}{5}.$

Let $CH\perp l, BH'\perp l.$ Lower diagram shows that

$\angle CAE = \angle ABH' = \alpha$ (perpendicular sides)

and $\angle CQE = 2\alpha$ (the same intersept $\overset{\Large\frown} {CE}).$

$$\tan\alpha = \frac {1}{8}, \sin2\alpha = \frac{2 \tan \alpha}{1 + \tan^2 \alpha} = \frac {16}{65}, \cos2\alpha = \frac{1 - \tan^2 \alpha}{1 + \tan^2 \alpha} = \frac {63}{65}.$$ The area

$$[ACE] = [AQC]+[CQE]– [AQE].$$ Hence

$$[ACE] =\frac{AQ^2}{2} \left(\sin 2\alpha + \sin 2\beta - \sin(2\alpha + 2\beta)\right),$$ $$[ACE] = 8\left( \frac{16}{65}+\frac{4}{5} - \frac{4}{5}\cdot \frac{63}{65} - \frac{3}{5}\cdot \frac{16}{65}\right) = \frac{64}{65}\implies \boxed {129}.$$ vladimir.shelomovskii@gmail.com, vvsss

Solution 10 (Similar Triangles, Angle Chasing, and Ptolemy's)

We begin by extending $\overline{AB}$ upwards until it intersects Circle $\mathcal{Q}$. We can call this point of intersection $F$. Connect $F$ with $E$, $C$, and $A$ for future use.

Create a trapezoid with points $B$, $C$, and the origins of Circles $\mathcal{P}$ and $\mathcal{Q}$. After quick inspection, we can conclude that the distance between the origins is 5 and that $\overline{BC}$ is 4.

(Note: It is important to understand that we could simply use the formula for the distance between the tangent points on a line and two different circles, which is $2\cdot \sqrt{a\cdot b}$, where $a$ and $b$ are the two respective radii of the circles. In our case, we get $2\cdot \sqrt{4} = 4$.

Using similar triangles or homotheties, $AE=4\cdot AD$ and $AF=4\cdot BA$. $BC^2 = BA\cdot BF$.

$$16 = BA\cdot (5\cdot BA)$$ $$BA = \frac{4}{\sqrt{5}}$$ $$AF = \frac{16}{\sqrt{5}}$$ $$BF = 4\cdot \sqrt{5}$$ Inspecting $\triangle{BFC}$, we recognize that it is a right triangle ($\angle{BCF} = 90$) as the final length ($\overline{FC}$) being 8 would allow for an $x-2x-x\sqrt{5}$ triangle. Hence, the diameter of circle $\mathcal{Q}$ = $CF$. This also means that $\angle{CAF} = \angle{CEF} = 90$.

From the fact that $\triangle{ABC}$ is a right triangle:

$$AB^2 + AC^2 = BC^2$$ $$AC = \frac{8}{\sqrt{5}}$$ (Note: We could have also used $\triangle{FAC}$.)

Our next step is to start angle chasing to find any other similar triangles or shared angles. Label the intersection between $\overline{FC}$ and $\overline{DE}$ as $M$. Label $\angle{CAE} = \angle{CFE} = \theta$. Since $\angle{BAC} = 90$, $\angle{EAF} = \angle{DAB} = 90-\theta$. Now, use the sine formula and the fact that the areas of $\triangle{DBA}$ and $\triangle{ACE}$ are equal to get:

$$\frac{1}{2}\cdot AC\cdot AE\cdot \sin{\theta} = \frac{1}{2}\cdot AD\cdot AB\cdot \sin{(90-\theta)}$$ Since $\sin{(90-\theta)} = \cos{\theta}$:

$$\tan{\theta} = \frac{AD}{2\cdot AE}$$ Using right $\triangle{FCE}$, since $\angle{ACM} = \angle{CFE}$, $\tan{\theta} = \frac{CE}{FE}$. Hence, plugging into the previous equation:

$$\frac{CE}{FE} = \frac{AD}{2\cdot AE}$$ Using the Pythagorean theorem on $\triangle{FCE}$, $FE = \sqrt{(64 - CE^2)}$. We also know that $AE=4\cdot AD$ Plugging back in:

$$\frac{CE}{\sqrt{(64 - CE^2)}} = \frac{AD}{2\cdot (4\cdot AD)}$$ $$\frac{CE}{\sqrt{(64 - CE^2)}} = \frac{1}{8}$$ . From here, we can square both sides and bring everything to one side to get:

$$EF^2 + 64EF - 64 = 0$$ .

$$EF = \frac{64}{\sqrt{65}}$$ $$CE = \frac{8}{\sqrt{65}}$$ We should also return to the fact that $\sin{\theta} = \frac{CE}{CF}$ from $\triangle{FCE}$, so

$$\sin{\theta} = \frac{1}{\sqrt{65}}$$ From the fact that $\angle{CAF} = \angle{CEF} = 90$, we can use Ptolemy's Theorem on quadrilateral $ACEF$. $AC\cdot EF + CE\cdot FA = CF\cdot AE$. Plugging in and solving, we get that $AE = \frac{16}{\sqrt{13}}$.

We now have all of our pieces to use the Sine Formula on $\triangle{ACE}$.

$$\frac{1}{2}\cdot AC\cdot AF\cdot \sin{\theta}$$ $$\frac{1}{2}\cdot \frac{8}{\sqrt{5}}\cdot \frac{16}{\sqrt{13}}\cdot \frac{1}{\sqrt{65}} = \frac{64}{65} = \boxed {129}$$ ~Solution by: armang32324

Solution 11 (Trig)

Let the center of the larger circle be $O,$ and the center of the smaller circle be $P.$ It is not hard to find the areas of $ACO$ and $ABP$ using pythagorean theorem, which are $\frac{32}{5}$ and $\frac{2}{5}$ respectively. Assign $\angle AOC=a,\angle COE=c,\angle DPB=d,\angle BPA=b.$ We can figure out that $\angle AOE=\angle DOA=\theta$ using vertical angles and isosceles triangles. Now, using $[ABC]=\frac{1}{2}ab\sin C$

$$[ACE]=[AOC]+[COE]-[AOC]=\dfrac{32}{5}-8\sin c-8\sin \theta,$$ $$[DAB]=[APD]+[APB]+[BPD]=\dfrac{2}{5}+\dfrac{1}{2}\sin \theta+\dfrac{1}{2}\sin d.$$ We can also figure out that $\sin a=\dfrac{4}{5},\cos a=\dfrac{3}{5},\sin b=\dfrac{4}{5}, \cos b=-\dfrac{3}{5}.$ Also, $c=\theta-a$ and $d=360-\theta-b.$ Using sum and difference identities:

$$\sin c=\dfrac{3}{5}\sin \theta-\dfrac{4}{5}\cos \theta,$$ $$\sin d=\dfrac{3}{5}\sin\theta-\dfrac{4}{5}\cos\theta.$$ (We can also notice that $c+d=360-\theta-b+\theta-a=360-(a+b)=180$ which means that $\sin c=\sin d.$) Substituting in the equations for $\sin c$ and $\sin d$ into the equations for $[ACE]$ and $[DAB],$ setting them equal, and simplifying:

$$3=2\sin\theta+3\cos\theta.$$ Solving this equation we get that $\sin\theta=\frac{12}{13}$ and $\cos\theta=\frac{5}{13}.$ Doing a lot of substitution gives us

$$[ACE]=[DAB]=\dfrac{64}{65},$$ which means the answer is $64+65=\boxed{129}.$

~BS2012

Solution 12 (Coordinates and Shoelace)

Let the centers of circles $\mathcal{P}$ and $\mathcal{Q}$ be $P$ and $Q$, respectively. Let $F$ be the point on $\overline{QC}$ such that $\overline{PF}$ is parallel to $\overline{BC}$. We find that $BC=PF=\sqrt{(4+1)^{2}-(4-1)^2}=4$.

Now, we define $A$ to be $(0,0)$. It follows that:

\begin{align} P &= \left(-\frac{4}{5},-\frac{3}{5}\right) \\ Q &= \left(\frac{16}{5},\frac{12}{5}\right) \\ B &= \left(-\frac{4}{5},-\frac{8}{5}\right) \\ C &= \left(\frac{16}{5},-\frac{8}{5}\right) \\ \end{align}

The equation of circle $P$ is $\left(x+\frac{4}{5}\right)^2+\left(y+\frac{3}{5}\right)^2 = 1^2$, which simplifies to $x^2+\frac{8}{5}x+y^2+\frac{6}{5}y=0$.

The equation of circle $Q$ is $\left(x-\frac{16}{5}\right)^2+\left(y-\frac{12}{5}\right)^2 = 4^2$, which simplifies to $x^2-\frac{32}{5}x+y^2-\frac{24}{5}y=0$.

Suppose points $D$ and $E$ lie on the line $y=mx$ for some slope $m$. We can now plug in $y=mx$ into the equations for the circles to find the coordinates of $D$ and $E$ in terms of $m$.

For point $D$:

$$x^2+\frac{8}{5}x+m^2x^2+\frac{6}{5}mx=0$$ $$(1+m^2)x^2+\left(\frac{8}{5}+\frac{6}{5}m\right)=0$$ Since $D \neq A$, we have $x \neq 0$, thus $x=\dfrac{-\left(\frac{8}{5}+\frac{6}{5}m\right)}{1+m^2}$.

For point $E$:

$$x^2-\frac{32}{5}x+m^2x^2-\frac{24}{5}mx=0$$ $$(1+m^2)x^2-\left(\frac{32}{5}+\frac{24}{5}m\right)=0$$ Since $E \neq A$, we have $x \neq 0$, thus $x=\dfrac{\left(\frac{32}{5}+\frac{24}{5}m\right)}{1+m^2}$.

We let $a=\dfrac{\left(\frac{8}{5}+\frac{6}{5}m\right)}{1+m^2}$ to write things quicker, and so the coordinates of $D$ are $(-a,-am)$ and the coordinates of $E$ are $(4a,4am)$.

Now, we can apply shoelace on $\triangle DBA$ and $\triangle ACE$: $[DBA] = \frac {1}{2}\left|(0 \cdot -\frac{8}{5} + -\frac{4}{5} \cdot -am + -a \cdot 0) - (0 \cdot -\frac{4}{5} + -\frac{8}{5} \cdot -a + -am \cdot 0)\right| = \dfrac{\left|-\frac{4}{5}am + \frac{8}{5}a\right|}{2} = \dfrac{\frac{8}{5}a-\frac{4}{5}am}{2}$.

$[ACE] = \frac {1}{2}\left|(0 \cdot -\frac{8}{5} + \frac{16}{5} \cdot 4am + 4a \cdot 0) - (0 \cdot \frac{16}{5} + -\frac{8}{5} \cdot 4a + 4am \cdot 0)\right| = \dfrac{\left|-\frac{32}{5}a-\frac{64}{5}am\right|}{2} = \dfrac{\frac{32}{5}a+\frac{64}{5}am}{2}$.

Note on determining the absolute value: We can assume $-\frac{1}{2} because$\overline{CA}$has a slope of$-\frac{1}{2}$and the area of$\triangle ACE$gets too big if$m>0$. So, we can also assume$a$ is positive.

Equating the two areas yields $\dfrac{\frac{8}{5}a-\frac{4}{5}am}{2} = \dfrac{\frac{32}{5}a+\frac{64}{5}am}{2} \implies -\frac{24}{5}a=\frac{68}{5}am \implies m=-\frac{6}{17}$.

We also know that $a=\dfrac{\left(\frac{8}{5}+\frac{6}{5}m\right)}{1+m^2}=\dfrac{\frac{8}{5}-\frac{36}{85}}{1+\frac{36}{289}}=\dfrac{\frac{100}{85}}{\frac{325}{289}}=\frac{68}{65}$.

Now, we can compute the area of $\triangle DBA$: $[DBA] = \dfrac{\frac{8}{5} \cdot \frac{68}{65} -\frac{4}{5} \cdot \frac{68}{65} \cdot \left(-\frac{6}{17}\right)}{2}=\dfrac{\frac{8\cdot68+4\cdot4\cdot6}{5\cdot65}}{2}=\frac{64}{65} \implies m+m=\boxed{129}$.

Solution 13 (Only Similar Triangles, No Trig)

Let $BH=x$, $HC = 4-x$

$x(4-x) = (\frac{8}{5})^2$ $x^2-4x+\frac{64}{25} = 0$ $25x^2-100x+64 - 0$ $(5x-16)(5x-4) = 0$

So $HC = \frac{16}{5}$, $BH = \frac{4}{5}$

$BC = \sqrt{(1+4)^2-(4-1)^2} = 4$ $\frac{r_1}{r_2} = \frac{AD}{AE} = \frac{1}{4} => \frac{h_1}{h_2} = 4$

Because $\triangle QIC \sim \triangle QGB$

$\frac{QC}{QB} = \frac{1}{4}$ $4QC = QB = QC+4$ $QC = \frac{4}{3}$

Because $O_1A = \frac{1}{5} O_2$

$AH = \frac{4}{5} O_1B = \frac{1}{5} O_2C = \frac{4}{5}+\frac{4}{5} = \frac{8}{5}$ $AQ = \sqrt{(\frac{8}{5})^2+(\frac{68}{15})^2} = \frac{4}{3}\sqrt{13}$ $CQ^2 = EQ\cdot AQ => EQ = \frac{\frac{16}{9}}{\frac{4}{3}\sqrt{13}} = \frac{4}{39}\sqrt{13}$ $AE = \frac{4}{3}\sqrt{13}-\frac{4}{39}\sqrt{13} = \frac{16}{13}\sqrt{13}$ $\sin{Q} = \frac{AH}{AQ} = \frac{\frac{8}{5}}{\frac{4}{3}\sqrt{13}} = \frac{6}{5\sqrt{13}}$ $CI = CQ\cdot \sin{Q} = \frac{4}{3}\frac{6}{5\sqrt{13}} = \frac{8}{5\sqrt{13}}$

So $S_{shaded} = \frac{1}{2}\cdot \frac{16}{13}\sqrt{13}\cdot \frac{8}{5\sqrt{13}} = \frac{64}{65}$

$64+65 = \boxed{129}$

-cassphe

Video Solution

https://youtu.be/LBCmQc2awMk

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