MathDB
Problems
Contests
National and Regional Contests
India Contests
India National Olympiad
2012 India National Olympiad
3
3
Part of
2012 India National Olympiad
Problems
(1)
Polys with int coefficients
Source: 2012 INMO (India National Olympiad), Problem #3
3/30/2016
Define a sequence
<
f
0
(
x
)
,
f
1
(
x
)
,
f
2
(
x
)
,
⋯
>
<f_0 (x), f_1 (x), f_2 (x), \dots>
<
f
0
(
x
)
,
f
1
(
x
)
,
f
2
(
x
)
,
⋯
>
of functions by
f
0
(
x
)
=
1
f_0 (x) = 1
f
0
(
x
)
=
1
f
1
(
x
)
=
x
f_1(x)=x
f
1
(
x
)
=
x
(
f
n
(
x
)
)
2
−
1
=
f
n
+
1
(
x
)
f
n
−
1
(
x
)
(f_n(x))^2 - 1 = f_{n+1}(x) f_{n-1}(x)
(
f
n
(
x
)
)
2
−
1
=
f
n
+
1
(
x
)
f
n
−
1
(
x
)
for
n
≥
1
n \ge 1
n
≥
1
. Prove that each
f
n
(
x
)
f_n (x)
f
n
(
x
)
is a polynomial with integer coefficients.
algebra