Difference between revisions of "2015 AIME I Problems/Problem 6"
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==Problem== | ==Problem== | ||
− | Point <math>A,B,C,D,</math> and <math>E</math> are equally spaced on a minor arc of a | + | Point <math>A,B,C,D,</math> and <math>E</math> are equally spaced on a minor arc of a circle. 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> | <asy> |
Revision as of 15:04, 11 August 2015
Problem
Point and
are equally spaced on a minor arc of a circle. 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
.
Solution
Let be the center of the circle with
on it.
Let and
.
is therefore
by way of circle
and
by way of circle
.
is
by way of circle
, and
is
by way of circle
.
This means that:
,
which when simplified yields , or
.
Since:
,
So:
is equal to
+
, which equates to
.
Plugging in yields
, or
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.