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

Revision as of 18:11, 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: $[asy] unitsize(1cm); draw((0,4*sqrt(3))--(8,4*sqrt(3))); draw((0,4*sqrt(3))--(4,0)); draw((8,4*sqrt(3))--(4,0)); draw((6,4*sqrt(3))--(4,0)); label("$x$",(5,2sqrt(3)),NNW); [/asy]$

@@ Line 6: / Line 6: @@
 <asy>
 unitsize(1cm);
-draw((0,3sqrt(3))--(3,0)--(12,0)--cycle);
+draw((0,4*sqrt(3))--(8,4*sqrt(3)));
-draw((0,5*sqrt(3))--(10,5*sqrt(3)));
+draw((0,4*sqrt(3))--(4,0));
-draw((0,5*sqrt(3))--(5,0));
+draw((8,4*sqrt(3))--(4,0));
-draw((10,5*sqrt(3))--(5,0));
+draw((6,4*sqrt(3))--(4,0));
-label("$A$",(0,3sqrt(3)),NNW);
+label("$x$",(5,2sqrt(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 18:11, 7 July 2019

Problem

Solution