Mock AIME 1 2010 Problems/Problem 13

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$",C1,NNW); dot(D); label("D",D,SW); dot(E); label("E",E,NW); [/asy]$

$\boxed{372}$ .

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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Mock AIME 1 2010 Problems/Problem 13

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