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National High School Mathematics League
1983 National High School Mathematics League
4
Two Simple Sets
Two Simple Sets
Source: 1983 National High School Mathematics League, Exam One, Problem 4
February 21, 2020
Problem Statement
Define two sets:
M
=
{
(
x
,
y
)
∣
y
≥
x
2
}
,
N
=
{
(
x
,
y
)
∣
x
2
+
(
y
−
a
)
2
≤
1
}
M=\{ (x,y)|y\geq x^2\} ,N=\{ (x,y)|x^2+(y-a)^2\leq 1\}
M
=
{(
x
,
y
)
∣
y
≥
x
2
}
,
N
=
{(
x
,
y
)
∣
x
2
+
(
y
−
a
)
2
≤
1
}
. If
M
∪
N
=
N
M\cup N=N
M
∪
N
=
N
, then the range value of
a
a
a
is
(A)
a
≥
1
1
4
(B)
a
=
1
1
4
(C)
a
≥
1
(D)
0
<
a
<
1
\text{(A)}a\geq 1\frac{1}{4}\qquad\text{(B)}a=1\frac{1}{4}\qquad\text{(C)}a\geq 1\qquad\text{(D)}0<a<1
(A)
a
≥
1
4
1
(B)
a
=
1
4
1
(C)
a
≥
1
(D)
0
<
a
<
1
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