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 1
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
denote the radius of the circle,
be the center of the circle. Then
. 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
and
condition we have
which gives
and
. 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
. Now
which gives
Now, we have the area for and the area for
, so we add them together:
The answer should therefore be . The answer is
.
-By SpecialBeing2017
Video Solution
Solution 2
Instead of considering the actual values of the areas, consider only the changes in the areas that result from moving point from the center of the circle. We will proceed by coordinates. Set the origin at the center of the circle and refer to the following diagram. Note that the area bounded by
and the arc
is fixed, so we only need to consider the relevant triangles.
Define one arbitrary unit as the distance that you need to move from
to change the area of
by
. We can see that
was moved down by
units to make the area defined by
,
, and
. Similarly,
was moved right by
to make the area defined by
,
, and
. This means that
has coordinates
.
Now, we need to consider how this displacement in affected the area defined by
,
, and
. This is equivalent to finding the shortest distance between
and the blue line in the diagram (as
and the blue line represents
while
is fixed). Using an isosceles right triangle, one can find the that shortest distance between
and this line is
.
Remembering the definition of our unit, this yields a final area of
-Archeon
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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