Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 6"

m (Solution)
(Solution)
Line 7: Line 7:
 
unitsize(1cm);
 
unitsize(1cm);
 
draw((0,3sqrt(3))--(3,0)--(12,0)--cycle);
 
draw((0,3sqrt(3))--(3,0)--(12,0)--cycle);
draw((0,12*sqrt(3))--(24,12*sqrt(3)));
+
draw((0,5*sqrt(3))--(10,5*sqrt(3)));
draw((0,12*sqrt(3))--(12,0));
+
draw((0,5*sqrt(3))--(5,0));
draw((3,0)--(120/19,2.46134));
+
draw((10,5*sqrt(3))--(5,0));
 
label("$A$",(0,3sqrt(3)),NNW);
 
label("$A$",(0,3sqrt(3)),NNW);
 
label("$B$",(3,0),SW);
 
label("$B$",(3,0),SW);

Revision as of 19:07, 7 July 2019

Problem

Three points $A,B,C$ are chosen at random on a circle. The probability that there exists a point $P$ inside an equilateral triangle $A_1B_1C_1$ such that $PA_1=BC,PB_1=AC,PC_1=AB$ can be expressed in the form $\frac{m} {n},$ where $m,n$ are relatively prime positive integers. Find $m+n.$

Solution

The problem asks for the probability that point $P$ is inside an equilateral triangle $A_1B_1C_1$. Let $x$, $y$, and $z$ be the three distances from point $P$ to each of the vertices, with $x$ being the longest distance. Let's consider the case in which point $P$ is actually on the line: [asy] unitsize(1cm); draw((0,3sqrt(3))--(3,0)--(12,0)--cycle); draw((0,5*sqrt(3))--(10,5*sqrt(3))); draw((0,5*sqrt(3))--(5,0)); draw((10,5*sqrt(3))--(5,0)); label("$A$",(0,3sqrt(3)),NNW); label("$B$",(3,0),SW); label("$C$",(12,0),ESE); label("$P$",(48/19,4.10223),NNE); label("$Q$",(120/19,2.46134),NE); label("$H$",(84/19,36sqrt(3)/19),NNE); [/asy]