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National and Regional Contests
China Contests
China Western Mathematical Olympiad
2017 China Western Mathematical Olympiad
5
5
Part of
2017 China Western Mathematical Olympiad
Problems
(1)
Find number of 9-tuples
Source: CWMI 2017 Q5
8/14/2017
Let
a
1
,
a
2
,
⋯
,
a
9
a_1,a_2,\cdots ,a_9
a
1
,
a
2
,
⋯
,
a
9
be
9
9
9
positive integers (not necessarily distinct) satisfying: for all
1
≤
i
<
j
<
k
≤
9
1\le i<j<k\le 9
1
≤
i
<
j
<
k
≤
9
, there exists
l
(
1
≤
l
≤
9
)
l (1\le l\le 9)
l
(
1
≤
l
≤
9
)
distinct from
i
,
j
i,j
i
,
j
and
j
j
j
such that
a
i
+
a
j
+
a
k
+
a
l
=
100
a_i+a_j+a_k+a_l=100
a
i
+
a
j
+
a
k
+
a
l
=
100
. Find the number of
9
9
9
-tuples
(
a
1
,
a
2
,
⋯
,
a
9
)
(a_1,a_2,\cdots ,a_9)
(
a
1
,
a
2
,
⋯
,
a
9
)
satisfying the above conditions.
algebra
combinatorics