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2020 Kürschák Competition
P2
P2
Part of
2020 Kürschák Competition
Problems
(1)
non-negative subadditive FE over the rationals
Source: 2020 Kürschák Competition P2
4/9/2021
Find all functions
f
:
Q
→
R
≥
0
f\colon \mathbb{Q}\to \mathbb{R}_{\geq 0}
f
:
Q
→
R
≥
0
such that for any two rational numbers
x
x
x
and
y
y
y
the following conditions hold[*]
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
f(x+y)\leq f(x)+f(y)
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
, [*]
f
(
x
y
)
=
f
(
x
)
f
(
y
)
f(xy)=f(x)f(y)
f
(
x
y
)
=
f
(
x
)
f
(
y
)
, [*]
f
(
2
)
=
1
/
2
f(2)=1/2
f
(
2
)
=
1/2
.
functional equation
algebra