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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2018 Korea Junior Math Olympiad
8
Evading function
Evading function
Source: KJMO 2018 p8
July 26, 2019
combinatorics
function
Problem Statement
For every set
S
S
S
with
n
(
≥
3
)
n(\ge3)
n
(
≥
3
)
distinct integers, show that there exists a function
f
:
{
1
,
2
,
…
,
n
}
→
S
f:\{1,2,\dots,n\}\rightarrow S
f
:
{
1
,
2
,
…
,
n
}
→
S
satisfying the following two conditions.(i)
{
f
(
1
)
,
f
(
2
)
,
…
,
f
(
n
)
}
=
S
\{ f(1),f(2),\dots,f(n)\} = S
{
f
(
1
)
,
f
(
2
)
,
…
,
f
(
n
)}
=
S
(ii)
2
f
(
j
)
≠
f
(
i
)
+
f
(
k
)
2f(j)\neq f(i)+f(k)
2
f
(
j
)
=
f
(
i
)
+
f
(
k
)
for all
1
≤
i
<
j
<
k
≤
n
1\le i<j<k\le n
1
≤
i
<
j
<
k
≤
n
.
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