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If is a differentiable function satisfying the condition
MATHEMATICS · Single correct · AoPS · Mathematics · Functions · Functions
Problem
If
f
(
x
)
f(x)
f
(
x
)
is a differentiable function satisfying the condition
f
(
100
x
)
=
x
+
f
(
100
x
−
100
)
f(100x) = x + f(100x - 100)
f
(
100
x
)
=
x
+
f
(
100
x
−
100
)
,
∀
x
∈
R
\forall x \in R
∀
x
∈
R
and
f
(
100
)
=
1
f(100) = 1
f
(
100
)
=
1
, then
f
(
10
n
)
f(10^n)
f
(
1
0
n
)
is (1) 5049 (2)
∑
r
=
1
100
r
\sum_{r=1}^{100} r
∑
r
=
1
100
r
(3)
∑
r
=
2
100
r
\sum_{r=2}^{100} r
∑
r
=
2
100
r
(4) 5050
Related theory
Functions
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