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Difference between revisions of "Mock AIME 1 2010 Problems/Problem 13"

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== Solution ==
 
== Solution ==
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<asy>
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 +
import geometry;
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size(8cm);
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point B = origin;
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point A = (3,8);
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point C = (12,0);
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triangle t = triangle(A,B,C);
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circle c = circumcircle(t);
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point B1 = foot(t.VB);
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point C1 = foot(t.VC);
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point D = intersectionpoint(line(B1,C1), line(B,C));
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pair[] e = intersectionpoints(line(A,D), c);
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point E = e[0];
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// Triangle ABC and Circumcircle
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draw(t);
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draw(c);
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// Altitudes
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draw(B--B1);
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draw(C--C1);
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// Segments AD,EB,BD, and B_1D
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draw(A--D);
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draw(E--B);
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draw(B--D);
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draw(B1--D);
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// Point Labels
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dot(A);
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label("A",A,NW);
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dot(B);
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label("B",B,SSW);
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dot(C);
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label("C",C,SE);
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dot(B1);
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label("B$_1$",B1,NE);
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dot(C1);
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label("C$_1$",C1,NNW);
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dot(D);
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label("D",D,SW);
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dot(E);
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label("E",E,NW);
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</asy>
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<math>\boxed{372}</math>.
 
<math>\boxed{372}</math>.
−
 
  
 
== See Also ==
 
== See Also ==
 
{{Mock AIME box|year=2010|n=1|num-b=12|num-a=14}}
 
{{Mock AIME box|year=2010|n=1|num-b=12|num-a=14}}
 
[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]

Revision as of 16:37, 11 August 2024

Problem

Suppose $\triangle ABC$ is inscribed in circle $\Gamma$. $B_1$ and $C_1$ are the feet of the altitude from $B$ to $CA$ and $C$ to $AB$, respectively. Let $D$ be the intersection of lines $\overline{B_1 C_1}$ and $\overline{BC}$, let $E$ be the point of intersection of $\Gamma$ and line $\overline{DA}$ distinct from $A$, and let $F$ be the foot of the perpendicular from $E$ to $BD$. Given that $BD = 28$, $EF = \frac{20 \sqrt{159}}{7}$, and $ED^2 + EB^2 = 3050$, and that $\tan m \angle ACB$ can be expressed in the form $\frac{a \sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers and $b$ is an integer not divisible by the square of any prime, find the last three digits of $a + b + c$.

Solution

[asy] import geometry; size(8cm); point B = origin; point A = (3,8); point C = (12,0); triangle t = triangle(A,B,C); circle c = circumcircle(t); point B1 = foot(t.VB); point C1 = foot(t.VC); point D = intersectionpoint(line(B1,C1), line(B,C)); pair[] e = intersectionpoints(line(A,D), c); point E = e[0]; // Triangle ABC and Circumcircle draw(t); draw(c); // Altitudes draw(B--B1); draw(C--C1); // Segments AD,EB,BD, and B_1D draw(A--D); draw(E--B); draw(B--D); draw(B1--D); // Point Labels dot(A); label("A",A,NW); dot(B); label("B",B,SSW); dot(C); label("C",C,SE); dot(B1); label("B$_1$",B1,NE); dot(C1); label("C$_1$",C1,NNW); dot(D); label("D",D,SW); dot(E); label("E",E,NW); [/asy]

$\boxed{372}$.

See Also

Mock AIME 1 2010 (Problems, Source)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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