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National and Regional Contests
Korea Contests
Korea National Olympiad
2014 Korea National Olympiad
4
function
function
Source: Korea National 2014 #8
January 21, 2015
function
algebra
ratio
equation
Problem Statement
Prove that there exists a function
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
that satisfies the following (1)
{
f
(
n
)
:
n
∈
N
}
\{f(n) : n\in\mathbb{N}\}
{
f
(
n
)
:
n
∈
N
}
is a finite set; and (2) For nonzero integers
x
1
,
x
2
,
…
,
x
1000
x_1, x_2, \ldots, x_{1000}
x
1
,
x
2
,
…
,
x
1000
that satisfy
f
(
∣
x
1
∣
)
=
f
(
∣
x
2
∣
)
=
⋯
=
f
(
∣
x
1000
∣
)
f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)
f
(
∣
x
1
∣
)
=
f
(
∣
x
2
∣
)
=
⋯
=
f
(
∣
x
1000
∣
)
, then
x
1
+
2
x
2
+
2
2
x
3
+
2
3
x
4
+
2
4
x
5
+
⋯
+
2
999
x
1000
≠
0
x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0
x
1
+
2
x
2
+
2
2
x
3
+
2
3
x
4
+
2
4
x
5
+
⋯
+
2
999
x
1000
=
0
.
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