Difference between revisions of "2015 AIME I Problems/Problem 6"
(Created page with "==Problem== Point <math>A,B,C,D,</math> and <math>E</math> are equally spaced on a minor arc of a cirle. Points <math>E,F,G,H,I</math> and <math>A</math> are equally spaced on...") |
(→Problem) |
||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
Point <math>A,B,C,D,</math> and <math>E</math> are equally spaced on a minor arc of a cirle. Points <math>E,F,G,H,I</math> and <math>A</math> are equally spaced on a minor arc of a second circle with center <math>C</math> as shown in the figure below. The angle <math>\angle ABD</math> exceeds <math>\angle AHG</math> by <math>12^\circ</math>. Find the degree measure of <math>\angle BAG</math>. | Point <math>A,B,C,D,</math> and <math>E</math> are equally spaced on a minor arc of a cirle. Points <math>E,F,G,H,I</math> and <math>A</math> are equally spaced on a minor arc of a second circle with center <math>C</math> as shown in the figure below. The angle <math>\angle ABD</math> exceeds <math>\angle AHG</math> by <math>12^\circ</math>. Find the degree measure of <math>\angle BAG</math>. | ||
| + | |||
| + | <asy> | ||
| + | pair A,B,C,D,E,F,G,H,I,O; | ||
| + | O=(0,0); | ||
| + | C=dir(90); | ||
| + | B=dir(70); | ||
| + | A=dir(50); | ||
| + | D=dir(110); | ||
| + | E=dir(130); | ||
| + | draw(arc(O,1,50,130)); | ||
| + | real x=2*sin(20*pi/180); | ||
| + | F=x*dir(228)+C; | ||
| + | G=x*dir(256)+C; | ||
| + | H=x*dir(284)+C; | ||
| + | I=x*dir(312)+C; | ||
| + | draw(arc(C,x,200,340)); | ||
| + | label("$A$",A,dir(0)); | ||
| + | label("$B$",B,dir(75)); | ||
| + | label("$C$",C,dir(90)); | ||
| + | label("$D$",D,dir(105)); | ||
| + | label("$E$",E,dir(180)); | ||
| + | label("$F$",F,dir(225)); | ||
| + | label("$G$",G,dir(260)); | ||
| + | label("$H$",H,dir(280)); | ||
| + | label("$I$",I,dir(315));</asy> | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2015|n=I|num-b=5|num-a=7}} | {{AIME box|year=2015|n=I|num-b=5|num-a=7}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 18:52, 20 March 2015
Problem
Point 
and 
are equally spaced on a minor arc of a cirle. Points 
and 
are equally spaced on a minor arc of a second circle with center 
as shown in the figure below. The angle 
exceeds 
by 
. Find the degree measure of 
.
![[asy] pair A,B,C,D,E,F,G,H,I,O; O=(0,0); C=dir(90); B=dir(70); A=dir(50); D=dir(110); E=dir(130); draw(arc(O,1,50,130)); real x=2*sin(20*pi/180); F=x*dir(228)+C; G=x*dir(256)+C; H=x*dir(284)+C; I=x*dir(312)+C; draw(arc(C,x,200,340)); label("$A$",A,dir(0)); label("$B$",B,dir(75)); label("$C$",C,dir(90)); label("$D$",D,dir(105)); label("$E$",E,dir(180)); label("$F$",F,dir(225)); label("$G$",G,dir(260)); label("$H$",H,dir(280)); label("$I$",I,dir(315));[/asy]](http://latex.artofproblemsolving.com/d/0/9/d09d2462dec1d60eaa6947ab799ce506fb914c99.png)
See Also
| 2015 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
