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National and Regional Contests
Romania Contests
Romania National Olympiad
1998 Romania National Olympiad
1
oh!! Integral
oh!! Integral
Source: Romania 1998
August 23, 2005
calculus
integration
function
real analysis
real analysis unsolved
Problem Statement
Suppose that
a
,
b
∈
R
+
a,b\in\mathbb{R}^+
a
,
b
∈
R
+
which
a
+
b
<
1
a+b<1
a
+
b
<
1
and
f
:
[
0
,
+
∞
)
→
[
0
,
+
∞
)
f:[0,+\infty) \rightarrow [0,+\infty)
f
:
[
0
,
+
∞
)
→
[
0
,
+
∞
)
be the increasing function s.t.
∀
x
≥
0
,
∫
0
x
f
(
t
)
d
t
=
∫
0
a
x
f
(
t
)
d
t
+
∫
0
b
x
f
(
t
)
d
t
\forall x\geq 0 ,\int _0^x f(t)dt=\int _0^{ax} f(t)dt+\int _0^{bx} f(t)dt
∀
x
≥
0
,
∫
0
x
f
(
t
)
d
t
=
∫
0
a
x
f
(
t
)
d
t
+
∫
0
b
x
f
(
t
)
d
t
. Prove that
∀
x
≥
0
,
f
(
x
)
=
0
\forall x\geq 0 , f(x)=0
∀
x
≥
0
,
f
(
x
)
=
0
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