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International Contests
Middle European Mathematical Olympiad
2023 Middle European Mathematical Olympiad
1
Functional inequality
Functional inequality
Source: MEMO 2023 I1
August 24, 2023
inequalities
MEMO2023
Problem Statement
For each pair
(
α
,
β
)
(\alpha, \beta)
(
α
,
β
)
of non-negative reals with
α
+
β
≥
2
\alpha+\beta \geq 2
α
+
β
≥
2
, determine all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
, such that
f
(
x
)
f
(
y
)
≤
f
(
x
y
)
+
α
x
+
β
y
f(x)f(y) \leq f(xy)+\alpha x+\beta y
f
(
x
)
f
(
y
)
≤
f
(
x
y
)
+
αx
+
β
y
for all reals
x
,
y
x, y
x
,
y
.
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