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2020 APMO
4
4
Part of
2020 APMO
Problems
(1)
Polynomial values contain some contiguous subsequence sum
Source: APMO 2020 Problem 4
6/9/2020
Let
Z
\mathbb{Z}
Z
denote the set of all integers. Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with integer coefficients that satisfy the following property:For any infinite sequence
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\dotsc
…
of integers in which each integer in
Z
\mathbb{Z}
Z
appears exactly once, there exist indices
i
<
j
i < j
i
<
j
and an integer
k
k
k
such that
a
i
+
a
i
+
1
+
⋯
+
a
j
=
P
(
k
)
a_i +a_{i+1} +\dotsb +a_j = P(k)
a
i
+
a
i
+
1
+
⋯
+
a
j
=
P
(
k
)
.
algebra
polynomial
APMO