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Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2022 Philippine MO
1
What's the difference?
What's the difference?
Source: Philippine MO 2022/1
March 18, 2022
functional equation
algebra
Problem Statement
Find all functions
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that
f
(
a
−
b
)
f
(
c
−
d
)
+
f
(
a
−
d
)
f
(
b
−
c
)
≤
(
a
−
c
)
f
(
b
−
d
)
f(a-b)f(c-d) + f(a-d)f(b-c) \leq (a-c)f(b-d)
f
(
a
−
b
)
f
(
c
−
d
)
+
f
(
a
−
d
)
f
(
b
−
c
)
≤
(
a
−
c
)
f
(
b
−
d
)
for all real numbers
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
and
d
d
d
.
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