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National and Regional Contests
Bosnia Herzegovina Contests
Regional Olympiad - Republic of Srpska
2004 Regional Olympiad - Republic of Srpska
1
polynomial and sequence
polynomial and sequence
Source: RS2004
March 20, 2005
algebra
polynomial
algebra proposed
Problem Statement
Define the sequence
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
by
a
1
=
1
a_1=1
a
1
=
1
,
a
2
=
p
a_2=p
a
2
=
p
and
a
n
+
1
=
p
a
n
−
a
n
−
1
for all
n
>
1.
a_{n+1}=pa_n-a_{n-1} \textrm { for all } n>1.
a
n
+
1
=
p
a
n
−
a
n
−
1
for all
n
>
1.
Prove that for
n
>
1
n>1
n
>
1
the polynomial
x
n
−
a
n
x
+
a
n
−
1
x^n-a_nx+a_{n-1}
x
n
−
a
n
x
+
a
n
−
1
is divisible by
x
2
−
p
x
+
1
x^2-px+1
x
2
−
p
x
+
1
. Using this result, solve the equation
x
4
−
56
x
+
15
=
0.
x^4-56x+15=0.
x
4
−
56
x
+
15
=
0.
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