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Putnam
2017 Putnam
A2
A2
Part of
2017 Putnam
Problems
(1)
Putnam 2017 A2
Source:
12/3/2017
Let
Q
0
(
x
)
=
1
Q_0(x)=1
Q
0
(
x
)
=
1
,
Q
1
(
x
)
=
x
,
Q_1(x)=x,
Q
1
(
x
)
=
x
,
and
Q
n
(
x
)
=
(
Q
n
−
1
(
x
)
)
2
−
1
Q
n
−
2
(
x
)
Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}
Q
n
(
x
)
=
Q
n
−
2
(
x
)
(
Q
n
−
1
(
x
)
)
2
−
1
for all
n
≥
2.
n\ge 2.
n
≥
2.
Show that, whenever
n
n
n
is a positive integer,
Q
n
(
x
)
Q_n(x)
Q
n
(
x
)
is equal to a polynomial with integer coefficients.
Putnam
Putnam 2017