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Problem
Problem
Consider a parabola
y
=
x
2
4
y = \frac{x^2}{4}
y
=
4
x
2
and the point
F
(
0
,
1
)
F(0, 1)
F
(
0
,
1
)
. Let
A
1
(
x
1
,
y
1
)
,
A
2
(
x
2
,
y
2
)
,
A
3
(
x
3
,
y
3
)
,
…
,
A
n
(
x
n
,
y
n
)
A_1(x_1, y_1), A_2(x_2, y_2), A_3(x_3, y_3), \dots, A_n(x_n, y_n)
A
1
(
x
1
,
y
1
)
,
A
2
(
x
2
,
y
2
)
,
A
3
(
x
3
,
y
3
)
,
…
,
A
n
(
x
n
,
y
n
)
are
n
n
n
points on the parabola such that
x
k
>
0
x_k > 0
x
k
>
0
and
∠
O
F
A
k
=
k
π
2
n
\angle OFA_k = \frac{k\pi}{2n}
∠
O
F
A
k
=
2
n
k
π
(
k
=
1
,
2
,
…
,
n
k = 1, 2, \dots, n
k
=
1
,
2
,
…
,
n
), If the value of
lim
n
→
∞
1
n
⋅
∑
k
=
1
n
F
A
k
=
m
π
\lim_{n \to \infty} \frac{1}{n} \cdot \sum_{k=1}^{n} FA_k = \frac{m}{\pi}
lim
n
→
∞
n
1
⋅
∑
k
=
1
n
F
A
k
=
π
m
, then
m
m
m
is .....
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