MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2023 Korea National Olympiad
3
3
Part of
2023 Korea National Olympiad
Problems
(1)
Actually, estimating the size of A is harder than the problem
Source: 2023 KMO P3
11/4/2023
For a given positive integer
n
(
≥
2
)
n(\ge 2)
n
(
≥
2
)
, find maximum positive integer
A
A
A
such that there exists
P
∈
Z
[
x
]
P \in \mathbb{Z}[x]
P
∈
Z
[
x
]
with degree
n
n
n
that satisfies the following two conditions.[*] For any
1
≤
k
≤
A
1 \le k \le A
1
≤
k
≤
A
, it satisfies that
A
∣
P
(
k
)
A \mid P(k)
A
∣
P
(
k
)
, and [*]
P
(
0
)
=
0
P(0)= 0
P
(
0
)
=
0
and the coefficient of the first term of
P
P
P
is
1
1
1
, which means that
P
(
x
)
P(x)
P
(
x
)
is in the following form where
c
2
,
c
3
,
⋯
,
c
n
c_2, c_3, \cdots, c_n
c
2
,
c
3
,
⋯
,
c
n
are all integers and
c
n
≠
0
c_n \neq 0
c
n
=
0
.
P
(
x
)
=
c
n
x
n
+
c
n
−
1
x
n
−
1
+
⋯
+
c
2
x
2
+
x
P(x) = c_nx^n + c_{n-1}x^{n-1}+\dots+c_2x^2+x
P
(
x
)
=
c
n
x
n
+
c
n
−
1
x
n
−
1
+
⋯
+
c
2
x
2
+
x
algebra
polynomial