2019 AIME II Problems/Problem 13
Problem
Regular octagon is inscribed in a circle of area
Point
lies inside the circle so that the region bounded by
and the minor arc
of the circle has area
while the region bounded by
and the minor arc
of the circle has area
There is a positive integer
such that the area of the region bounded by
and the minor arc
of the circle is equal to
Find
Solution
This problem is not difficult, but the calculation is tormenting.
The actual size of the diagram doesn't matter.
To make calculation easier, we discard the original area of the circle, , and assume the side length of the octagon is
Let denotes the radius of the circle,
be the center of the circle.
Now, we need to find the "D"shape, the small area enclosed by one side of the octagon and 1/8 of the circumference of the circle
Let
be the height of
,
be the height of
,
be the height of
,
From the 1/7 and 1/9 condition
we have
which gives
Now, let intersects
at
,
intersects
at
,
intersects
at
Clearly, is an isosceles right triangle, with right angle at
and the height with regard to which shall be
That is a common sense
which gives
Now, we have the area for and the area for
we add them together
The answer should therefore be
The final answer is, therefore,
-By SpecialBeing2017
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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