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Iran RMM TST
2019 Iran RMM TST
2
Nice polynomial with gcd greater than 1
Nice polynomial with gcd greater than 1
Source: Iran RMM TST 2019,day1 p2
July 30, 2019
polynomial
algebra
chebyshev polynomial
Problem Statement
Let
n
>
1
n >1
n
>
1
be a natural number and
T
n
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
a
n
−
2
x
n
−
2
+
.
.
.
+
a
1
x
1
+
a
0
T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0
T
n
(
x
)
=
x
n
+
a
n
−
1
x
n
−
1
+
a
n
−
2
x
n
−
2
+
...
+
a
1
x
1
+
a
0
.\\ Assume that for each nonzero real number
t
t
t
we have
T
n
(
t
+
1
t
)
=
t
n
+
1
t
n
T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n}
T
n
(
t
+
t
1
)
=
t
n
+
t
n
1
.\\ Prove that for each
0
≤
i
≤
n
−
1
0\le i \le n-1
0
≤
i
≤
n
−
1
g
c
d
(
a
i
,
n
)
>
1
gcd (a_i,n) >1
g
c
d
(
a
i
,
n
)
>
1
.Proposed by Morteza Saghafian
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