Difference between revisions of "2016 AIME I Problems/Problem 15"

Revision as of 14:48, 16 October 2016

Problem

Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$ . Line $\ell$ is tangent to $\omega_1$ and $\omega_2$ at $A$ and $B$ , respectively, with line $AB$ closer to point $X$ than to $Y$ . Circle $\omega$ passes through $A$ and $B$ intersecting $\omega_1$ again at $D \neq A$ and intersecting $\omega_2$ again at $C \neq B$ . The three points $C$ , $Y$ , $D$ are collinear, $XC = 67$ , $XY = 47$ , and $XD = 37$ . Find $AB^2$ .

Solution

Solution 1

By the Radical Axis Theorem $AD, XY, BC$ concur at point $E$ .

Let $AB$ and $EY$ intersect at $S$ . Note that because $AXDY$ and $CYXB$ are cyclic, by Miquel's Theorem $AXBE$ is cyclic as well. Thus $\angle AEX = \angle ABX = \angle XCB = \angle XYB$ and $\angle XEB = \angle XAB = \angle XDA = \angle XYA.$ Thus $AY \parallel EB$ and $YB \parallel EA$ , so $AEBY$ is a parallelogram. Hence $AS = SB$ and $SE = SY$ . But notice that $DXE$ and $EXC$ are similar by $AA$ Similarity, so $XE^2 = XD \cdot XC = 37 \cdot 67$ . But $XE^2 - XY^2 = (XE + XY)(XE - XY) = EY \cdot 2XS = 2SY \cdot 2SX = 4SA^2 = AB^2.$ Hence $AB^2 = 37 \cdot 67 - 47^2 = \boxed{270}.$

Solution 2

First, we note that as $\triangle XDY$ and $\triangle XYC$ have bases along the same line, $\frac{[\triangle XDY]}{[\triangle XYC]}=\frac{DY}{YC}$ . We can also find the ratio of their areas using the circumradius area formula. If $R_1$ is the radius of $\omega_1$ and if $R_2$ is the radius of $\omega_2$ , then $\frac{[\triangle XDY]}{[\triangle XYC]}=\frac{(37\cdot 47\cdot DY)/(4R_1)}{(47\cdot 67\cdot YC)/(4R_2)}=\frac{37\cdot DY\cdot R_2}{67\cdot YC\cdot R_1}.$ Since we showed this to be $\frac{DY}{YC}$ , we see that $\frac{R_2}{R_1}=\frac{67}{37}$ .

We extend $AD$ and $BC$ to meet at point $P$ , and we extend $AB$ and $CD$ to meet at point $Q$ as shown below. $[asy] size(200); import olympiad; real R1=45,R2=67*R1/37; real m1=sqrt(R1^2-23.5^2); real m2=sqrt(R2^2-23.5^2); pair o1=(0,0),o2=(m1+m2,0),x=(m1,23.5),y=(m1,-23.5); draw(circle(o1,R1)); draw(circle(o2,R2)); pair q=(-R1/(R2-R1)*o2.x,0); pair a=tangent(q,o1,R1,2); pair b=tangent(q,o2,R2,2); pair d=intersectionpoints(circle(o1,R1),q--y+15*(y-q))[0]; pair c=intersectionpoints(circle(o2,R2),q--y+15*(y-q))[1]; pair p=extension(a,d,b,c); dot(q^^a^^b^^x^^y^^c^^d^^p); draw(q--b^^q--c); draw(p--d^^p--c^^x--y); draw(a--y^^b--y); draw(d--x--c); label("$A$",a,NW,fontsize(8)); label("$B$",b,NE,fontsize(8)); label("$C$",c,SE,fontsize(8)); label("$D$",d,SW,fontsize(8)); label("$X$",x,2*WNW,fontsize(8)); label("$Y$",y,3*S,fontsize(8)); label("$P$",p,N,fontsize(8)); label("$Q$",q,W,fontsize(8)); [/asy]$ As $ABCD$ is cyclic, we know that $\angle BCD=180-\angle DAB=\angle BAP$ . But then as $AB$ is tangent to $\omega_2$ at $B$ , we see that $\angle BCD=\angle ABY$ . Therefore, $\angle ABY=\angle BAP$ , and $BY\parallel PD$ . A similar argument shows $AY\parallel PC$ . These parallel lines show $\triangle PDC\sim\triangle ADY\sim\triangle BYC$ . Also, we showed that $\frac{R_2}{R_1}=\frac{67}{37}$ , so the ratio of similarity between $\triangle ADY$ and $\triangle BYC$ is $\frac{37}{67}$ , or rather $\frac{AD}{BY}=\frac{DY}{YC}=\frac{YA}{CB}=\frac{37}{67}.$ We can now use the parallel lines to find more similar triangles. As $\triangle AQD\sim \triangle BQY$ , we know that $\frac{QA}{QB}=\frac{QD}{QY}=\frac{AD}{BY}=\frac{37}{67}.$ Setting $QA=37x$ , we see that $QB=67x$ , hence $AB=30x$ , and the problem simplifies to finding $30^2x^2$ . Setting $QD=37^2y$ , we also see that $QY=37\cdot 67y$ , hence $DY=37\cdot 30y$ . Also, as $\triangle AQY\sim \triangle BQC$ , we find that $\frac{QY}{QC}=\frac{YA}{CB}=\frac{37}{67}.$ As $QY=37\cdot 67y$ , we see that $QC=67^2y$ , hence $YC=67\cdot30y$ .

Applying Power of a Point to point $Q$ with respect to $\omega_2$ , we find $67^2x^2=37\cdot 67^3 y^2,$ or $x^2=37\cdot 67 y^2$ . We wish to find $AB^2=30^2x^2=30^2\cdot 37\cdot 67y^2$ .

Applying Stewart's Theorem to $\triangle XDC$ , we find $37^2\cdot (67\cdot 30y)+67^2\cdot(37\cdot 30y)=(67\cdot 30y)\cdot (37\cdot 30y)\cdot (104\cdot 30y)+47^2\cdot (104\cdot 30y).$ We can cancel $30\cdot 104\cdot y$ from both sides, finding $37\cdot 67=30^2\cdot 67\cdot 37y^2+47^2$ . Therefore, $AB^2=30^2\cdot 37\cdot 67y^2=37\cdot 67-47^2=\boxed{270}.$

Solution 3

$[asy] size(9cm); import olympiad; real R1=45,R2=67*R1/37; real m1=sqrt(R1^2-23.5^2); real m2=sqrt(R2^2-23.5^2); pair o1=(0,0),o2=(m1+m2,0),x=(m1,23.5),y=(m1,-23.5); draw(circle(o1,R1)); draw(circle(o2,R2)); pair q=(-R1/(R2-R1)*o2.x,0); pair a=tangent(q,o1,R1,2); pair b=tangent(q,o2,R2,2); pair d=intersectionpoints(circle(o1,R1),q--y+15*(y-q))[0]; pair c=intersectionpoints(circle(o2,R2),q--y+15*(y-q))[1]; dot(a^^b^^x^^y^^c^^d); draw(x--y); draw(a--y^^b--y); draw(d--x--c); draw(a--b--c--d--cycle); draw(x--a^^x--b); label("$A$",a,NW,fontsize(9)); label("$B$",b,NE,fontsize(9)); label("$C$",c,SE,fontsize(9)); label("$D$",d,SW,fontsize(9)); label("$X$",x,2*N,fontsize(9)); label("$Y$",y,3*S,fontsize(9)); [/asy]$ First of all, since quadrilaterals $ADYX$ and $XYCB$ are cyclic, we can let $\angle DAX = \angle XYC = \theta$ , and $\angle XYD = \angle CBX = 180 - \theta$ , due to the properties of cyclic quadrilaterals. In addition, let $\angle BAX = x$ and $\angle ABX = y$ . Then, since quadrilateral $ABCD$ is cyclic as well, we have the following sums: $\theta + x +\angle XCY + y = 180^{\circ}$ $180^{\circ} - \theta + y + \angle XDY + x = 180^{\circ}$ Cancelling out $180^{\circ}$ in the second equation isolating $\theta$ yields $\theta = y + \angle XDY + x$ . Substituting $\theta$ back into the first equation, we obtain $2x + 2y + \angle XCY + \angle XDY = 180^{\circ}$ Since $x + y +\angle XAY + \angle XCY + \angle DAY = 180^{\circ}$ $x + y + \angle XDY + \angle XCY + \angle DAY = 180^{\circ}$ we can then imply that $\angle DAY = x + y$ . Similarly, $\angle YBC = x + y$ . So then $\angle DXY = \angle YXC = x + y$ , so since we know that $XY$ bisects $\angle DXC$ , we can solve for $DY$ and $YC$ with Stewart’s Theorem. Let $DY = 37n$ and $YC = 67n$ . Then $37n \cdot 67n \cdot 104n + 47^2 \cdot 104n = 37^2 \cdot 67n + 67^2 \cdot 37n$ $37n \cdot 67n + 47^2 = 37 \cdot 67$ $n^2 = \frac{270}{2479}$ Now, since $\angle AYX = x$ and $\angle BYX = y$ , $\angle AYB = x + y$ . From there, let $\angle AYD = \alpha$ and $\angle BYC = \beta$ . From angle chasing we can derive that $\angle YDX = \angle YAX = \beta - x$ and $\angle YCX = \angle YBX = \alpha - y$ . From there, since $\angle ADX = x$ , it is quite clear that $\angle ADY = \beta$ , and $\angle YAB = \beta$ can be found similarly. From there, since $\angle ADY = \angle YAB = \angle BYC = \beta$ and $\angle DAY = \angle AYB = \angle YBC = x + y$ , we have $AA$ similarity between triangles $DAY$ , $AYB$ , and $YBC$ . Therefore the length of $AY$ is the geometric mean of the lengths of $DA$ and $YB$ (from $\triangle DAY \sim \triangle AYB$ ). However, $\triangle DAY \sim \triangle AYB \sim \triangle YBC$ yields the proportion $\frac{AD}{DY} = \frac{YA}{AB} = \frac{BY}{YC}$ ; therefore, the length of $AB$ is the geometric mean of the lengths of $DY$ and $YC$ . We can now simply use arithmetic to calculate $AB^2$ . $AB^2 = DY \cdot YC$ $AB^2 = 37 \cdot 67 \cdot \frac{270}{2479}$ $AB^2 = \boxed{270}$

-Solution by TheBoomBox77

@@ Line 105: / Line 105: @@
 <cmath>37n \cdot 67n + 47^2 = 37 \cdot 67</cmath>
 <cmath>n^2 = \frac{270}{2479}</cmath>
-Now, since <math>\angle AYX = x</math> and <math>\angle BYX = y</math>, <math>\angle AYB = x + y</math>. From there, let <math>\angle AYD = \alpha</math> and <math>\angle BYC = \beta</math>. From angle chasing we can derive that <math>\angle YDX = \angle YAX = \beta - x</math> and <math>\angle YCX = \angle YBX = \alpha - y</math>. Now it is clear that <math>\triangle DAY \sim \triangle AYB \sim \triangle YBC</math>. Therefore the length of <math>AY</math> is the geometric mean of the lengths of <math>DA</math> and <math>YB</math> (from <math>\triangle DAY \sim \triangle AYB</math>). However, due to the proportion <math>\frac{AD}{DY} = \frac{YA}{AB} = \frac{BY}{YC}</math>, the length of <math>AB</math> is the geometric mean of the lengths of <math>DY</math> and <math>YC</math>.
+Now, since <math>\angle AYX = x</math> and <math>\angle BYX = y</math>, <math>\angle AYB = x + y</math>. From there, let <math>\angle AYD = \alpha</math> and <math>\angle BYC = \beta</math>. From angle chasing we can derive that <math>\angle YDX = \angle YAX = \beta - x</math> and <math>\angle YCX = \angle YBX = \alpha - y</math>. From there, since <math>\angle ADX = x</math>, it is quite clear that <math>\angle ADY = \beta</math>, and <math>\angle YAB = \beta</math> can be found similarly. From there, since <math>\angle ADY = \angle YAB = \angle BYC = \beta</math> and <math>\angle DAY = \angle AYB = \angle YBC = x + y</math>, we have <math>AA</math> similarity between triangles <math>DAY</math>, <math>AYB</math>, and <math>YBC</math>. Therefore the length of <math>AY</math> is the geometric mean of the lengths of <math>DA</math> and <math>YB</math> (from <math>\triangle DAY \sim \triangle AYB</math>). However, <math>\triangle DAY \sim \triangle AYB \sim \triangle YBC</math> yields the proportion <math>\frac{AD}{DY} = \frac{YA}{AB} = \frac{BY}{YC}</math>; therefore, the length of <math>AB</math> is the geometric mean of the lengths of <math>DY</math> and <math>YC</math>.
 We can now simply use arithmetic to calculate <math>AB^2</math>.
 <cmath>AB^2 = DY \cdot YC</cmath>

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