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Problems
Contests
National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1951 Moscow Mathematical Olympiad
207
207
Part of
1951 Moscow Mathematical Olympiad
Problems
(1)
MMO 207 Moscow MO 1951 bus route, 14 stops, 25 passengers
Source:
8/7/2019
* A bus route has
14
14
14
stops (counting the first and the last). A bus cannot carry more than
25
25
25
passengers. We assume that a passenger takes a bus from
A
A
A
to
B
B
B
if (s)he enters the bus at
A
A
A
and gets off at
B
B
B
. Prove that for any bus route: a) there are
8
8
8
distinct stops
A
1
,
B
1
,
A
2
,
B
2
,
A
3
,
B
3
,
A
4
,
B
4
A_1, B_1, A_2, B_2, A_3, B_3, A_4, B_4
A
1
,
B
1
,
A
2
,
B
2
,
A
3
,
B
3
,
A
4
,
B
4
such that no passenger rides from
A
k
A_k
A
k
to
B
k
B_k
B
k
for all
k
=
1
,
2
,
3
,
4
k = 1, 2, 3, 4
k
=
1
,
2
,
3
,
4
(#)b) there might not exist
10
10
10
distinct stops
A
1
,
B
1
,
.
.
.
,
A
5
,
B
5
A_1, B_1, . . . , A_5, B_5
A
1
,
B
1
,
...
,
A
5
,
B
5
with property (#).
combinatorics