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2022 Math Prize for Girls Problems
19
19
Part of
2022 Math Prize for Girls Problems
Problems
(1)
Math Prize 2022 Problem 19
Source:
10/12/2022
Let
S
−
S_-
S
−
be the semicircular arc defined by
(
x
+
1
)
2
+
(
y
−
3
2
)
2
=
1
4
and
x
≤
−
1.
(x + 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \le -1.
(
x
+
1
)
2
+
(
y
−
2
3
)
2
=
4
1
and
x
≤
−
1.
Let
S
+
S_+
S
+
be the semicircular arc defined by
(
x
−
1
)
2
+
(
y
−
3
2
)
2
=
1
4
and
x
≥
1.
(x - 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \ge 1.
(
x
−
1
)
2
+
(
y
−
2
3
)
2
=
4
1
and
x
≥
1.
Let
R
R
R
be the locus of points
P
P
P
such that
P
P
P
is the intersection of two lines, one of the form
A
x
+
B
y
=
1
Ax + By = 1
A
x
+
B
y
=
1
where
(
A
,
B
)
∈
S
−
(A, B) \in S_-
(
A
,
B
)
∈
S
−
and the other of the form
C
x
+
D
y
=
1
Cx + Dy = 1
C
x
+
Dy
=
1
where
(
C
,
D
)
∈
S
+
(C, D) \in S_+
(
C
,
D
)
∈
S
+
. What is the area of
R
R
R
?