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National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2022 Flanders Math Olympiad
3
3
Part of
2022 Flanders Math Olympiad
Problems
(1)
Arne has 2n + 1 tickets
Source: Flanders Math Olympiad 2022 p3
12/24/2022
Arne has
2
n
+
1
2n + 1
2
n
+
1
tickets. Each card has one number on it. One card has the number
0
0
0
on it. The natural numbers
1
,
2
,
.
.
.
,
n
1, 2, . . . , n
1
,
2
,
...
,
n
occur on exactly two cards each. Prove that Arne can arrange cards in a row so that there are exactly
m
m
m
cards between the two cards with the number
m
m
m
, for every
m
ā
{
1
,
2
,
.
.
.
,
n
}
m \in \{1, 2, . . . , n\}
m
ā
{
1
,
2
,
...
,
n
}
.
combinatorics