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Serbia Contests
Serbia Team Selection Test
1982 Yugoslav Team Selection Test
Problem 2
polynomials of a certain form
polynomials of a certain form
Source: Yugoslav TST 1982 P2
May 29, 2021
algebra
polynomial
Problem Statement
Find all polynomials
P
n
(
x
)
P_n(x)
P
n
(
x
)
of the form
P
n
(
x
)
=
n
!
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
1
x
+
(
−
1
)
n
n
(
n
+
1
)
,
P_n(x)=n!x^n+a_{n-1}x^{n-1}+\ldots+a_1x+(-1)^nn(n+1),
P
n
(
x
)
=
n
!
x
n
+
a
n
−
1
x
n
−
1
+
…
+
a
1
x
+
(
−
1
)
n
n
(
n
+
1
)
,
with integer coefficients, such that its roots
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
satisfy
k
≤
x
k
≤
k
+
1
k\le x_k\le k+1
k
≤
x
k
≤
k
+
1
for
k
=
1
,
2
,
…
,
n
k=1,2,\ldots,n
k
=
1
,
2
,
…
,
n
.
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