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2
Problem 8 IMC 2004 Macedonia
Problem 8 IMC 2004 Macedonia
Source:
July 26, 2004
function
integration
geometry
perimeter
inequalities
IMC
college contests
Problem Statement
Let
f
,
g
:
[
a
,
b
]
→
[
0
,
∞
)
f,g:[a,b]\to [0,\infty)
f
,
g
:
[
a
,
b
]
→
[
0
,
∞
)
be two continuous and non-decreasing functions such that each
x
∈
[
a
,
b
]
x\in [a,b]
x
∈
[
a
,
b
]
we have
∫
a
x
f
(
t
)
d
t
≤
∫
a
x
g
(
t
)
d
t
and
∫
a
b
f
(
t
)
d
t
=
∫
a
b
g
(
t
)
d
t
.
\int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt.
∫
a
x
f
(
t
)
d
t
≤
∫
a
x
g
(
t
)
d
t
and
∫
a
b
f
(
t
)
d
t
=
∫
a
b
g
(
t
)
d
t
.
Prove that
∫
a
b
1
+
f
(
t
)
d
t
≥
∫
a
b
1
+
g
(
t
)
d
t
.
\int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt.
∫
a
b
1
+
f
(
t
)
d
t
≥
∫
a
b
1
+
g
(
t
)
d
t
.
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