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Switzerland Contests
Switzerland Team Selection Test
2001 Switzerland Team Selection Test
6
6
Part of
2001 Switzerland Team Selection Test
Problems
(1)
f(x+y) >= f(x)+ f(y), prove f(x) \le 2x for all x \in [0,1]
Source: Switzerland - Swiss TST 2001 p6
2/18/2020
A function
f
:
[
0
,
1
]
→
R
f : [0,1] \to R
f
:
[
0
,
1
]
→
R
has the following properties: (a)
f
(
x
)
≥
0
f(x) \ge 0
f
(
x
)
≥
0
for
0
<
x
<
1
0 < x < 1
0
<
x
<
1
, (b)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
, (c)
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
f(x+y) \ge f(x)+ f(y)
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
whenever
x
,
y
,
x
+
y
∈
[
0
,
1
]
x,y,x+y \in [0,1]
x
,
y
,
x
+
y
∈
[
0
,
1
]
. Prove that
f
(
x
)
≤
2
x
f(x) \le 2x
f
(
x
)
≤
2
x
for all
x
∈
[
0
,
1
]
x \in [0,1]
x
∈
[
0
,
1
]
.
function
Functional inequality
inequalities
algebra