MathDB
Problems
Contests
National and Regional Contests
USA Contests
USA - College-Hosted Events
Harvard-MIT Mathematics Tournament
2021 Harvard-MIT Mathematics Tournament.
4
2021 Team #4
2021 Team #4
Source:
June 27, 2021
Summation
combinatorics
Problem Statement
Let
k
k
k
and
n
n
n
be positive integers and let
S
=
{
(
a
1
,
…
,
a
k
)
∈
Z
k
∣
0
≤
a
k
≤
⋯
≤
a
1
≤
n
,
a
1
+
⋯
+
a
k
=
n
}
S=\{(a_1,\ldots,a_k)\in \mathbb{Z}^{k}\;|\; 0\leq a_k\leq\cdots\leq a_1 \leq n,a_1+\cdots+a_k=n\}
S
=
{(
a
1
,
…
,
a
k
)
∈
Z
k
∣
0
≤
a
k
≤
⋯
≤
a
1
≤
n
,
a
1
+
⋯
+
a
k
=
n
}
. Determine, with proof, the value of
∑
(
a
1
,
…
,
a
k
)
∈
S
(
n
a
1
)
(
a
1
a
2
)
⋯
(
a
k
−
1
a
k
)
\sum_{(a_1,\ldots,a_k)\in S}\binom{n}{a_1}\binom{a_1}{a_2}\cdots\binom{a_{k-1}}{a_k}
(
a
1
,
…
,
a
k
)
∈
S
∑
(
a
1
n
)
(
a
2
a
1
)
⋯
(
a
k
a
k
−
1
)
in terms of
k
k
k
and
n
n
n
, where the sum is over all
k
k
k
-tuples in
S
S
S
.
Back to Problems
View on AoPS