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

Revision as of 18:04, 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: $[asy] unitsize(1cm); draw((0,3sqrt(3))--(3,0)--(12,0)--cycle); draw((3,0)--(84/19,36sqrt(3)/19)); draw((3,0)--(48/19, 4.10223)); draw((3,0)--(120/19,2.46134)); label("$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$",(84/19,36sqrt(3)/19),NNE); [/asy]$

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

Revision as of 18:04, 7 July 2019

Problem

Solution

@@ Line 7: / Line 7: @@
 unitsize(1cm);
 draw((0,3sqrt(3))--(3,0)--(12,0)--cycle);
-draw((0, 18*sqrt(3))--(18, 0));
+draw((3,0)--(84/19,36sqrt(3)/19));
-draw((18, 0)--(36, 18*sqrt(3)));
+draw((3,0)--(48/19, 4.10223));
+draw((3,0)--(120/19,2.46134));
+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>