Difference between revisions of "2019 AIME I Problems/Problem 13"
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− | The | + | ==Problem 13== |
+ | Triangle <math>ABC</math> has side lengths <math>AB=4</math>, <math>BC=5</math>, and <math>CA=6</math>. Points <math>D</math> and <math>E</math> are on ray <math>AB</math> with <math>AB<AD<AE</math>. The point <math>F \neq C</math> is a point of intersection of the circumcircles of <math>\triangle ACD</math> and <math>\triangle EBC</math> satisfying <math>DF=2</math> and <math>EF=7</math>. Then <math>BE</math> can be expressed as <math>\tfrac{a+b\sqrt{c}}{d}</math>, where <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are positive integers such that <math>a</math> and <math>d</math> are relatively prime, and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c+d</math>. | ||
+ | |||
+ | ==Solution 1== | ||
+ | <asy> | ||
+ | unitsize(20); | ||
+ | pair A, B, C, D, E, F, X, O1, O2; | ||
+ | A = (0, 0); B = (4, 0); | ||
+ | C = intersectionpoints(circle(A, 6), circle(B, 5))[0]; | ||
+ | D = B + (5/4 * (1 + sqrt(2)), 0); E = D + (4 * sqrt(2), 0); | ||
+ | F = intersectionpoints(circle(D, 2), circle(E, 7))[1]; | ||
+ | X = extension(A, E, C, F); | ||
+ | O1 = circumcenter(C, A, D); | ||
+ | O2 = circumcenter(C, B, E); | ||
+ | |||
+ | filldraw(A--B--C--cycle, lightcyan, deepcyan); | ||
+ | filldraw(D--E--F--cycle, lightmagenta, deepmagenta); | ||
+ | draw(B--D, gray(0.6)); | ||
+ | draw(C--F, gray(0.6)); | ||
+ | draw(circumcircle(C, A, D), dashed); | ||
+ | draw(circumcircle(C, B, E), dashed); | ||
+ | |||
+ | dot("$A$", A, dir(A-O1)); | ||
+ | dot("$B$", B, dir(240)); | ||
+ | dot("$C$", C, dir(120)); | ||
+ | dot("$D$", D, dir(40)); | ||
+ | dot("$E$", E, dir(E-O2)); | ||
+ | dot("$F$", F, dir(270)); | ||
+ | dot("$X$", X, dir(140)); | ||
+ | |||
+ | label("$6$", (C+A)/2, dir(C-A)*I, deepcyan); | ||
+ | label("$5$", (C+B)/2, dir(B-C)*I, deepcyan); | ||
+ | label("$4$", (A+B)/2, dir(A-B)*I, deepcyan); | ||
+ | label("$7$", (F+E)/2, dir(F-E)*I, deepmagenta); | ||
+ | label("$2$", (F+D)/2, dir(D-F)*I, deepmagenta); | ||
+ | label("$4\sqrt{2}$", (D+E)/2, dir(E-D)*I, deepmagenta); | ||
+ | label("$a$", (B+X)/2, dir(B-X)*I, gray(0.3)); | ||
+ | label("$a\sqrt{2}$", (D+X)/2, dir(D-X)*I, gray(0.3)); | ||
+ | </asy> | ||
+ | |||
+ | 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>. | ||
+ | |||
+ | (Solution by TheUltimate123) | ||
+ | |||
+ | ==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>. | ||
+ | |||
+ | Because of exterior angles, | ||
+ | |||
+ | <math>\angle ACB = \angle CBE - \angle CAD</math> | ||
+ | |||
+ | But <math>\angle CBE = \angle CFE</math> because <math>CBFE</math> is cyclic. In addition, <math>\angle CAD = \angle CFD</math> because <math>CAFD</math> is cyclic. Therefore, <math>\angle ACB = \angle CFE - \angle CFD</math>. But <math>\angle CFE - \angle CFD = \angle DFE</math>, so <math>\angle ACB = \angle DFE</math>. Using Law of Cosines on <math>\triangle ABC</math>, we can figure out that <math>\cos(\angle ACB) = \frac{3}{4}</math>. Since <math>\angle ACB = \angle DFE</math>, <math>\cos(\angle DFE) = \frac{3}{4}</math>. We are given that <math>DF = 2</math> and <math>FE = 7</math>, so we can use Law of Cosines on <math>\triangle DEF</math> to find that <math>DE = 4\sqrt{2}</math>. | ||
+ | |||
+ | Let <math>G</math> be the intersection of segment <math>\overline{AE}</math> and <math>\overline{CF}</math>. Using Power of a Point with respect to <math>G</math> within <math>\omega_1</math>, we find that <math>AG \cdot GD = CG \cdot GF</math>. We can also apply Power of a Point with respect to <math>G</math> within <math>\omega_2</math> to find that <math>CG \cdot GF = BG \cdot GE</math>. Therefore, <math>AG \cdot GD = BG \cdot GE</math>. | ||
+ | |||
+ | <math>AG \cdot GD = BG \cdot GE</math> | ||
+ | |||
+ | <math>(AB + BG) \cdot GD = BG \cdot (GD + DE)</math> | ||
+ | |||
+ | <math>AB \cdot GD + BG \cdot GD = BG \cdot GD + BG \cdot DE</math> | ||
+ | |||
+ | <math>AB \cdot GD = BG \cdot DE</math> | ||
+ | |||
+ | <math>4 \cdot GD = BG \cdot 4\sqrt{2}</math> | ||
+ | |||
+ | <math>GD = BG \cdot \sqrt{2}</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 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 that <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 | ||
+ | ==Solution 4 (No <C = <DFE, no LoC)== | ||
+ | Let <math>P=AE\cap CF</math>. Let <math>CP=5x</math> and <math>BP=5y</math>; from <math>\triangle{CBP}\sim\triangle{EFP}</math> we have <math>EP=7x</math> and <math>FP=7y</math>. From <math>\triangle{CAP}\sim\triangle{DFP}</math> we have <math>\frac{6}{4+5y}=\frac{2}{7y}</math> giving <math>y=\frac{1}{4}</math>. So <math>BP=\frac{5}{4}</math> and <math>FP=\frac{7}{4}</math>. These similar triangles also gives us <math>DP=\frac{5}{3}x</math> so <math>DE=\frac{16}{3}x</math>. Now, Stewart's Theorem on <math>\triangle{FEP}</math> and cevian <math>FD</math> tells us that <cmath>\frac{560}{9}x^3+28x=\frac{49}{3}x+\frac{245}{3}x,</cmath>so <math>x=\frac{3\sqrt{2}}{4}</math>. Then <math>BE=\frac{5}{4}+7x=\frac{5+21\sqrt{2}}{4}</math> so the answer is <math>\boxed{032}</math> as desired. (Solution by Trumpeter, but not added to the Wiki by Trumpeter) | ||
− | |||
− | |||
==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=I|num-b=12|num-a=14}} | {{AIME box|year=2019|n=I|num-b=12|num-a=14}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:26, 2 January 2021
Contents
Problem 13
Triangle has side lengths , , and . Points and are on ray with . The point is a point of intersection of the circumcircles of and satisfying and . Then can be expressed as , where , , , and are positive integers such that and are relatively prime, and is not divisible by the square of any prime. Find .
Solution 1
Notice that By the Law of Cosines, Then, Let , , and . Then, However, since , , but since , and the requested sum is .
(Solution by TheUltimate123)
Solution 2
Define to be the circumcircle of and to be the circumcircle of .
Because of exterior angles,
But because is cyclic. In addition, because is cyclic. Therefore, . But , so . Using Law of Cosines on , we can figure out that . Since , . We are given that and , so we can use Law of Cosines on to find that .
Let be the intersection of segment and . Using Power of a Point with respect to within , we find that . We can also apply Power of a Point with respect to within to find that . Therefore, .
Note that is similar to . . Also note that is similar to , which gives us . Solving this system of linear equations, we get . Now, we can solve for , which is equal to . This simplifies to , which means our answer is .
Solution 3
Construct and let . Let . Using , Using , it can be found that This also means that . It suffices to find . It is easy to see the following: Using reverse Law of Cosines on , . Using Law of Cosines on gives , so . -franchester
Solution 4 (No <C = <DFE, no LoC)
Let . Let and ; from we have and . From we have giving . So and . These similar triangles also gives us so . Now, Stewart's Theorem on and cevian tells us that so . Then so the answer is as desired. (Solution by Trumpeter, but not added to the Wiki by Trumpeter)
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.