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Putnam
2004 Putnam
A3
Putnam 2004 A3
Putnam 2004 A3
Source:
December 11, 2004
Putnam
linear algebra
matrix
induction
college contests
Problem Statement
Define a sequence
{
u
n
}
n
=
0
∞
\{u_n\}_{n=0}^{\infty}
{
u
n
}
n
=
0
∞
by
u
0
=
u
1
=
u
2
=
1
,
u_0=u_1=u_2=1,
u
0
=
u
1
=
u
2
=
1
,
and thereafter by the condition that
det
∣
u
n
u
n
+
1
u
n
+
2
u
n
+
3
∣
=
n
!
\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!
det
u
n
u
n
+
2
u
n
+
1
u
n
+
3
=
n
!
for all
n
≥
0.
n\ge 0.
n
≥
0.
Show that
u
n
u_n
u
n
is an integer for all
n
.
n.
n
.
(By convention,
0
!
=
1
0!=1
0
!
=
1
.)
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