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Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2016 Latvia Baltic Way TST
8
8
Part of
2016 Latvia Baltic Way TST
Problems
(1)
3n - 2 participants in chess festival
Source: 2016 Latvia BW TST P8
12/17/2022
3
n
−
2
3n - 2
3
n
−
2
participants took part in the chess festival, some of them played one game of chess with each other. Prove that at least one of the following statements holds:(A) One can find
n
n
n
chess players
A
1
,
A
2
,
.
.
.
,
A
n
A_1 , A_2 , . . . , A_n
A
1
,
A
2
,
...
,
A
n
suchthat Ai has played a game with
A
i
+
1
A_{i+1}
A
i
+
1
for all
i
=
1
,
.
.
.
,
n
−
1
i = 1, ...,n -1
i
=
1
,
...
,
n
−
1
.(B) Seven chess players can be found in
B
1
,
.
.
.
,
B
7
B_1 , . . . , B_7
B
1
,
...
,
B
7
, who have not played with each other, except perhaps three pairs
(
B
1
,
B
2
)
(B_1, B_2)
(
B
1
,
B
2
)
,
(
B
3
,
B
4
)
(B_3, B_4)
(
B
3
,
B
4
)
and
(
B
5
,
B
6
)
(B_5, B_6)
(
B
5
,
B
6
)
, each of whom may or may not have played a game of chess.
combinatorics