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2017 Macedonia National Olympiad
Problem 1
Problem 1
Part of
2017 Macedonia National Olympiad
Problems
(1)
Macedonia National Olympiad 2017 Problem 1
Source: Macedonia National Olympiad 2017
4/8/2017
Find all functions
f
:
N
→
N
f:\mathbb{N} \to \mathbb{N}
f
:
N
→
N
such that for each natural integer
n
>
1
n>1
n
>
1
and for all
x
,
y
∈
N
x,y \in \mathbb{N}
x
,
y
∈
N
the following holds:
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
∑
k
=
1
n
−
1
(
n
k
)
x
n
−
k
y
k
f(x+y) = f(x) + f(y) + \sum_{k=1}^{n-1} \binom{n}{k}x^{n-k}y^k
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
k
=
1
∑
n
−
1
(
k
n
)
x
n
−
k
y
k
function
algebra