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2015 IMO Shortlist
A5
Integer to odds functional equation
Integer to odds functional equation
Source: 2015 ISL A5
July 7, 2016
function
algebra
functional equation
IMO Shortlist
Problem Statement
Let
2
Z
+
1
2\mathbb{Z} + 1
2
Z
+
1
denote the set of odd integers. Find all functions
f
:
Z
↦
2
Z
+
1
f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1
f
:
Z
↦
2
Z
+
1
satisfying
f
(
x
+
f
(
x
)
+
y
)
+
f
(
x
−
f
(
x
)
−
y
)
=
f
(
x
+
y
)
+
f
(
x
−
y
)
f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y)
f
(
x
+
f
(
x
)
+
y
)
+
f
(
x
−
f
(
x
)
−
y
)
=
f
(
x
+
y
)
+
f
(
x
−
y
)
for every
x
,
y
∈
Z
x, y \in \mathbb{Z}
x
,
y
∈
Z
.
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