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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2014 Korea Junior Math Olympiad
8
8
Part of
2014 Korea Junior Math Olympiad
Problems
(1)
n students in m clubs
Source: KJMO 2014 p8
5/2/2019
Let there be
n
n
n
students and
m
m
m
clubs. The students joined the clubs so that the following is true: - For all students
x
x
x
, you can choose some clubs such that
x
x
x
is the only student who joined all of the chosen clubs. Let the number of clubs each student joined be
a
1
,
a
2
,
.
.
.
,
a
m
a_1,a_2,...,a_m
a
1
,
a
2
,
...
,
a
m
. Prove that
a
1
!
(
m
−
a
1
)
!
+
a
2
!
(
m
−
a
2
)
!
+
.
.
.
+
a
n
!
(
m
−
a
n
)
!
≤
m
!
a_1!(m - a_1)! + a_2!(m - a_2)! + ... + a_n!(m -a_n)! \le m!
a
1
!
(
m
−
a
1
)!
+
a
2
!
(
m
−
a
2
)!
+
...
+
a
n
!
(
m
−
a
n
)!
≤
m
!
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