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Serbia Contests
Federal Math Competition of Serbia and Montenegro
2003 Federal Math Competition of S&M
Problem 2
A functional inequality
A functional inequality
Source:
October 11, 2016
functional equation
Functional inequality
inequalities
Problem Statement
Let
f
:
[
0
,
1
]
→
R
f : [0, 1] \to\ R
f
:
[
0
,
1
]
→
R
be a function such that :-
1.
)
1.)
1.
)
f
(
x
)
≥
0
f(x) \ge 0
f
(
x
)
≥
0
for all
x
x
x
in
[
0
,
1
]
[0,1]
[
0
,
1
]
.
2.
)
2.)
2.
)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
.
3.
)
3.)
3.
)
If
x
1
,
x
2
x_1 , x_2
x
1
,
x
2
are in
[
0
,
1
]
[0,1]
[
0
,
1
]
such that
x
1
+
x
2
≤
1
x_1 + x_2 \le 1
x
1
+
x
2
≤
1
, then
f
(
x
1
)
+
f
(
x
2
)
≤
f
(
x
1
+
x
2
)
f(x_1) + f(x_2) \le f(x_1 + x_2)
f
(
x
1
)
+
f
(
x
2
)
≤
f
(
x
1
+
x
2
)
. Show that
f
(
x
)
≤
2
x
f(x) \le 2x
f
(
x
)
≤
2
x
for all
x
x
x
in
[
0
,
1
]
[0,1]
[
0
,
1
]
.
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