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National and Regional Contests
PEN Problems
PEN K Problems
32
K 32
K 32
Source:
May 25, 2007
function
Functional Equations
Problem Statement
Find all functions
f
:
Z
2
→
R
+
f: \mathbb{Z}^{2}\to \mathbb{R}^{+}
f
:
Z
2
→
R
+
such that for all
i
,
j
∈
Z
i, j \in \mathbb{Z}
i
,
j
∈
Z
:
f
(
i
,
j
)
=
f
(
i
+
1
,
j
)
+
f
(
i
,
j
+
1
)
+
f
(
i
−
1
,
j
)
+
f
(
i
,
j
−
1
)
4
.
f(i,j)=\frac{f(i+1, j)+f(i,j+1)+f(i-1,j)+f(i,j-1)}{4}.
f
(
i
,
j
)
=
4
f
(
i
+
1
,
j
)
+
f
(
i
,
j
+
1
)
+
f
(
i
−
1
,
j
)
+
f
(
i
,
j
−
1
)
.
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