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Putnam
1997 Putnam
6
Putnam 1997 A6
Putnam 1997 A6
Source:
May 30, 2014
Putnam
college contests
Problem Statement
For a positive integer
n
n
n
and any real number
c
c
c
, define
x
k
x_k
x
k
recursively by :
x
0
=
0
,
x
1
=
1
and for
k
≥
0
,
x
k
+
2
=
c
x
k
+
1
−
(
n
−
k
)
x
k
k
+
1
x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}
x
0
=
0
,
x
1
=
1
and for
k
≥
0
,
x
k
+
2
=
k
+
1
c
x
k
+
1
−
(
n
−
k
)
x
k
Fix
n
n
n
and then take
c
c
c
to be the largest value for which
x
n
+
1
=
0
x_{n+1}=0
x
n
+
1
=
0
. Find
x
k
x_k
x
k
in terms of
n
n
n
and
k
,
1
≤
k
≤
n
k,\; 1\le k\le n
k
,
1
≤
k
≤
n
.
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