Difference between revisions of "2019 AIME I Problems/Problem 13"

(Solution 2)
(Note: I put my solution as Solution 1 as I feel like it has a diagram and is formatted well.)
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==Solution 1==
 
==Solution 1==
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<asy>
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size(10cm);
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pair A, B, C, D, EE, F, X;
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B=dir(270-aCos(9/16));
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C=dir(270+aCos(9/16));
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A=intersectionpoint(circle((0, 0), 1), (B+0.01*(1, 3sqrt(7))) -- (B+100*(1, 3sqrt(7))));
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D=B-5/16*(sqrt(2)+1)*(A-B);
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EE=B-(5+21*sqrt(2))/16*(A-B);
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F=intersectionpoints(circumcircle(A, C, D), circumcircle(B, C, EE))[0];
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X=extension(A, B, C, F);
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draw(B -- C -- A -- EE -- F -- C); draw(D -- F);
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draw(circumcircle(A, C, D)); draw(circumcircle(C, EE, F));
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dot("$A$", A, N);
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dot("$B$", B, NW);
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dot("$C$", C, E);
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dot("$D$", D, SW);
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dot("$E$", EE, SW);
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dot("$F$", F, W);
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</asy>
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Notice that <cmath>\angle DFE=\angle CFE-\angle CFD=\angle CBE-\angle CAD=180-B-A=C.</cmath>By the Law of Cosines, <cmath>\cos C=\frac{AC^2+BC^2-AB^2}{2\cdot AC\cdot BC}=\frac34.</cmath>Then, <cmath>DE^2=DF^2+EF^2-2\cdot DF\cdot EF\cos C=32\implies DE=4\sqrt2.</cmath>Let <math>X=\overline{AB}\cap\overline{CF}</math>, <math>a=XB</math>, and <math>b=XD</math>. Then, <cmath>XA\cdot XD=XC\cdot XF=XB\cdot XE\implies b(a+4)=a(b+4\sqrt2)\implies b=a\sqrt2.</cmath>However, since <math>\triangle XFD\sim\triangle XAC</math>, <math>XF=\tfrac{4+a}3</math>, but since <math>\triangle XFE\sim\triangle XBC</math>, <cmath>\frac75=\frac{4+a}{3a}\implies a=\frac54\implies BE=a+a\sqrt2+4\sqrt2=\frac{5+21\sqrt2}4,</cmath>and the requested sum is <math>5+21+2+4=\boxed{032}</math>.
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(Solution by TheUltimate123)
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 +
==Solution 2==
  
 
Define <math>\omega_1</math> to be the circumcircle of <math>\triangle ACD</math> and <math>\omega_2</math> to be the circumcircle of <math>\triangle EBC</math>.
 
Define <math>\omega_1</math> to be the circumcircle of <math>\triangle ACD</math> and <math>\omega_2</math> to be the circumcircle of <math>\triangle EBC</math>.
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Note that <math>\triangle GAC</math> is similar to <math>\triangle GFD</math>. <math>GF = \frac{BG + 4}{3}</math>. Also note that <math>\triangle GBC</math> is similar to <math>\triangle GFE</math>, which gives us <math>GF = \frac{7 \cdot BG}{5}</math>. Solving this system of linear equations, we get <math>BG = \frac{5}{4}</math>. Now, we can solve for <math>BE</math>, which is equal to <math>BG(\sqrt{2} + 1) + 4\sqrt{2}</math>. This simplifies to <math>\frac{5 + 21\sqrt{2}}{4}</math>, which means our answer is <math>\boxed{032}</math>.
 
Note that <math>\triangle GAC</math> is similar to <math>\triangle GFD</math>. <math>GF = \frac{BG + 4}{3}</math>. Also note that <math>\triangle GBC</math> is similar to <math>\triangle GFE</math>, which gives us <math>GF = \frac{7 \cdot BG}{5}</math>. Solving this system of linear equations, we get <math>BG = \frac{5}{4}</math>. Now, we can solve for <math>BE</math>, which is equal to <math>BG(\sqrt{2} + 1) + 4\sqrt{2}</math>. This simplifies to <math>\frac{5 + 21\sqrt{2}}{4}</math>, which means our answer is <math>\boxed{032}</math>.
  
==Solution 2==
+
==Solution 3==
 
Construct <math>FC</math> and let <math>FC\cap AE=K</math>. Let <math>FK=x</math>. Using <math>\triangle FKE\sim \triangle BKC</math>, <cmath>BK=\frac{5}{7}x</cmath> Using <math>\triangle FDK\sim ACK</math>, it can be found taht <cmath>3x=AK=4+\frac{5}{7}x\to x=\frac{7}{4}</cmath> This also means that <math>BK=\frac{21}{4}-4=\frac{5}{4}</math>. It suffices to find <math>KE</math>. It is easy to see the following: <cmath>180-\angle ABC=\angle KBC=\angle KFE</cmath> Using reverse Law of Cosines on <math>\triangle ABC</math>, <math>\cos{\angle ABC}=\frac{1}{8}\to \cos{180-\angle ABC}=\frac{-1}{8}</math>. Using Law of Cosines on <math>\triangle EFK</math> gives <math>KE=\frac{21\sqrt 2}{4}</math>, so <math>BE=\frac{5+21\sqrt 2}{4}\to \textbf{032}</math>.
 
Construct <math>FC</math> and let <math>FC\cap AE=K</math>. Let <math>FK=x</math>. Using <math>\triangle FKE\sim \triangle BKC</math>, <cmath>BK=\frac{5}{7}x</cmath> Using <math>\triangle FDK\sim ACK</math>, it can be found taht <cmath>3x=AK=4+\frac{5}{7}x\to x=\frac{7}{4}</cmath> This also means that <math>BK=\frac{21}{4}-4=\frac{5}{4}</math>. It suffices to find <math>KE</math>. It is easy to see the following: <cmath>180-\angle ABC=\angle KBC=\angle KFE</cmath> Using reverse Law of Cosines on <math>\triangle ABC</math>, <math>\cos{\angle ABC}=\frac{1}{8}\to \cos{180-\angle ABC}=\frac{-1}{8}</math>. Using Law of Cosines on <math>\triangle EFK</math> gives <math>KE=\frac{21\sqrt 2}{4}</math>, so <math>BE=\frac{5+21\sqrt 2}{4}\to \textbf{032}</math>.
 
-franchester
 
-franchester

Revision as of 19:00, 15 March 2019

Problem 13

Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

Solution 1

[asy] size(10cm); pair A, B, C, D, EE, F, X; B=dir(270-aCos(9/16)); C=dir(270+aCos(9/16)); A=intersectionpoint(circle((0, 0), 1), (B+0.01*(1, 3sqrt(7))) -- (B+100*(1, 3sqrt(7)))); D=B-5/16*(sqrt(2)+1)*(A-B); EE=B-(5+21*sqrt(2))/16*(A-B); F=intersectionpoints(circumcircle(A, C, D), circumcircle(B, C, EE))[0]; X=extension(A, B, C, F);  draw(B -- C -- A -- EE -- F -- C); draw(D -- F); draw(circumcircle(A, C, D)); draw(circumcircle(C, EE, F));  dot("$A$", A, N); dot("$B$", B, NW); dot("$C$", C, E); dot("$D$", D, SW); dot("$E$", EE, SW); dot("$F$", F, W); [/asy] Notice that \[\angle DFE=\angle CFE-\angle CFD=\angle CBE-\angle CAD=180-B-A=C.\]By the Law of Cosines, \[\cos C=\frac{AC^2+BC^2-AB^2}{2\cdot AC\cdot BC}=\frac34.\]Then, \[DE^2=DF^2+EF^2-2\cdot DF\cdot EF\cos C=32\implies DE=4\sqrt2.\]Let $X=\overline{AB}\cap\overline{CF}$, $a=XB$, and $b=XD$. Then, \[XA\cdot XD=XC\cdot XF=XB\cdot XE\implies b(a+4)=a(b+4\sqrt2)\implies b=a\sqrt2.\]However, since $\triangle XFD\sim\triangle XAC$, $XF=\tfrac{4+a}3$, but since $\triangle XFE\sim\triangle XBC$, \[\frac75=\frac{4+a}{3a}\implies a=\frac54\implies BE=a+a\sqrt2+4\sqrt2=\frac{5+21\sqrt2}4,\]and the requested sum is $5+21+2+4=\boxed{032}$.

(Solution by TheUltimate123)

Solution 2

Define $\omega_1$ to be the circumcircle of $\triangle ACD$ and $\omega_2$ to be the circumcircle of $\triangle EBC$.

Because of exterior angles,

$\angle ACB = \angle CBE - \angle CAD$

But $\angle CBE = \angle CFE$ because $CBFE$ is cyclic. In addition, $\angle CAD = \angle CFD$ because $CAFD$ is cyclic. Therefore, $\angle ACB = \angle CFE - \angle CFD$. But $\angle CFE - \angle CFD = \angle DFE$, so $\angle ACB = \angle DFE$. Using Law of Cosines on $\triangle ABC$, we can figure out that $\cos(\angle ACB) = \frac{3}{4}$. Since $\angle ACB = \angle DFE$, $\cos(\angle DFE) = \frac{3}{4}$. We are given that $DF = 2$ and $FE = 7$, so we can use Law of Cosines on $\triangle DEF$ to find that $DE = 4\sqrt{2}$.

Let $G$ be the intersection of segment $\overline{AE}$ and $\overline{CF}$. Using Power of a Point with respect to $G$ within $\omega_1$, we find that $AG \cdot GD = CG \cdot GF$. We can also apply Power of a Point with respect to $G$ within $\omega_2$ to find that $CG \cdot GF = BG \cdot GE$. Therefore, $AG \cdot GD = BG \cdot GE$.

$AG \cdot GD = BG \cdot GE$

$(AB + BG) \cdot GD = BG \cdot (GD + DE)$

$AB \cdot GD + BG \cdot GD = BG \cdot GD + BG \cdot DE$

$AB \cdot GD = BG \cdot DE$

$4 \cdot GD = BG \cdot 4\sqrt{2}$

$GD = BG \cdot \sqrt{2}$

Note that $\triangle GAC$ is similar to $\triangle GFD$. $GF = \frac{BG + 4}{3}$. Also note that $\triangle GBC$ is similar to $\triangle GFE$, which gives us $GF = \frac{7 \cdot BG}{5}$. Solving this system of linear equations, we get $BG = \frac{5}{4}$. Now, we can solve for $BE$, which is equal to $BG(\sqrt{2} + 1) + 4\sqrt{2}$. This simplifies to $\frac{5 + 21\sqrt{2}}{4}$, which means our answer is $\boxed{032}$.

Solution 3

Construct $FC$ and let $FC\cap AE=K$. Let $FK=x$. Using $\triangle FKE\sim \triangle BKC$, \[BK=\frac{5}{7}x\] Using $\triangle FDK\sim ACK$, it can be found taht \[3x=AK=4+\frac{5}{7}x\to x=\frac{7}{4}\] This also means that $BK=\frac{21}{4}-4=\frac{5}{4}$. It suffices to find $KE$. It is easy to see the following: \[180-\angle ABC=\angle KBC=\angle KFE\] Using reverse Law of Cosines on $\triangle ABC$, $\cos{\angle ABC}=\frac{1}{8}\to \cos{180-\angle ABC}=\frac{-1}{8}$. Using Law of Cosines on $\triangle EFK$ gives $KE=\frac{21\sqrt 2}{4}$, so $BE=\frac{5+21\sqrt 2}{4}\to \textbf{032}$. -franchester

See Also

2019 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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