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IMO Shortlist
2017 IMO Shortlist
A8
Shortlist 2017/A8
Shortlist 2017/A8
Source:
July 10, 2018
algebra
function
IMO Shortlist
Problem Statement
A function
f
:
R
→
R
f:\mathbb{R} \to \mathbb{R}
f
:
R
→
R
has the following property:
For every
x
,
y
∈
R
such that
(
f
(
x
)
+
y
)
(
f
(
y
)
+
x
)
>
0
,
we have
f
(
x
)
+
y
=
f
(
y
)
+
x
.
\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.
For every
x
,
y
∈
R
such that
(
f
(
x
)
+
y
)
(
f
(
y
)
+
x
)
>
0
,
we have
f
(
x
)
+
y
=
f
(
y
)
+
x
.
Prove that
f
(
x
)
+
y
≤
f
(
y
)
+
x
f(x)+y \leq f(y)+x
f
(
x
)
+
y
≤
f
(
y
)
+
x
whenever
x
>
y
x>y
x
>
y
.
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