Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(created solution page)
 
(diagram)
Line 3: Line 3:
  
 
== Solution ==
 
== 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>
 +
 
<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
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Mock_AIME_1_2010_Problems/Problem_13&oldid=224940"