2015 AIME I Problems/Problem 6
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
.
Solution
Let O be the center of the circle with ABCDE on it. Let x=ED=DC=CB=BA and y=EF=FG=GH=HI=IA.
is therefore 5y by way of circle C and 180-2x by way of circle O.
is 180-3x/2 by way of circle O, and
is 180-3y/2 by way of circle C.
This means that 180-3x/2=180-3y/2+12, which when simplified yields 3x/2+12=3y/2, or x+8=y.
Since 5y=180-2x,
5x+40=180-2x
7x=140
x=20
y=28.
is equal to
+
, which equates to 3x/2+y.
Plugging in yields 30+28, or 058
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 |
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