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APMO
1994 APMO
1
Functional equation
Functional equation
Source: APMO 1994
March 11, 2006
function
induction
algebra unsolved
algebra
Problem Statement
Let
f
:
R
→
R
f: \Bbb{R} \rightarrow \Bbb{R}
f
:
R
→
R
be a function such that (i) For all
x
,
y
∈
R
x,y \in \Bbb{R}
x
,
y
∈
R
,
f
(
x
)
+
f
(
y
)
+
1
≥
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y)
f
(
x
)
+
f
(
y
)
+
1
≥
f
(
x
+
y
)
≥
f
(
x
)
+
f
(
y
)
(ii) For all
x
∈
[
0
,
1
)
x \in [0,1)
x
∈
[
0
,
1
)
,
f
(
0
)
≥
f
(
x
)
f(0) \geq f(x)
f
(
0
)
≥
f
(
x
)
, (iii)
−
f
(
−
1
)
=
f
(
1
)
=
1
-f(-1) = f(1) = 1
−
f
(
−
1
)
=
f
(
1
)
=
1
. Find all such functions
f
f
f
.
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