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National and Regional Contests
The Philippines Contests
Philippine MO
2024 Philippine Math Olympiad
P1
P1
Part of
2024 Philippine Math Olympiad
Problems
(1)
A two-variable functional equation
Source: Philippine Mathematical Olympiad 2024 P1
2/24/2024
Let
f
:
Z
2
→
Z
f:\mathbb{Z}^2\rightarrow\mathbb{Z}
f
:
Z
2
→
Z
be a function satisfying
f
(
x
+
1
,
y
)
+
f
(
x
,
y
+
1
)
+
1
=
f
(
x
,
y
)
+
f
(
x
+
1
,
y
+
1
)
f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1)
f
(
x
+
1
,
y
)
+
f
(
x
,
y
+
1
)
+
1
=
f
(
x
,
y
)
+
f
(
x
+
1
,
y
+
1
)
for all integers
x
x
x
and
y
y
y
. Can it happen that
∣
f
(
x
,
y
)
∣
≤
2024
|f(x,y)|\leq 2024
∣
f
(
x
,
y
)
∣
≤
2024
for all
x
,
y
∈
Z
x,y\in\mathbb{Z}
x
,
y
∈
Z
?
algebra
functional equation