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2016 IMO Shortlist
A7
Strange FE with max
Strange FE with max
Source: 2016 IMO Shortlist A7
July 19, 2017
IMO Shortlist
functional equation
algebra
Problem Statement
Find all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that
f
(
0
)
≠
0
f(0)\neq 0
f
(
0
)
=
0
and for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
,
f
(
x
+
y
)
2
=
2
f
(
x
)
f
(
y
)
+
max
{
f
(
x
2
+
y
2
)
,
f
(
x
2
)
+
f
(
y
2
)
}
.
f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}.
f
(
x
+
y
)
2
=
2
f
(
x
)
f
(
y
)
+
max
{
f
(
x
2
+
y
2
)
,
f
(
x
2
)
+
f
(
y
2
)
}
.
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