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Problems
Contests
International Contests
IMO Shortlist
1983 IMO Shortlist
16
Set of polynomials
Set of polynomials
Source:
September 9, 2010
algebra
polynomial
Sequence
coefficients
IMO Shortlist
Problem Statement
Let
F
(
n
)
F(n)
F
(
n
)
be the set of polynomials
P
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
P(x) = a_0+a_1x+\cdots+a_nx^n
P
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
, with
a
0
,
a
1
,
.
.
.
,
a
n
∈
R
a_0, a_1, . . . , a_n \in \mathbb R
a
0
,
a
1
,
...
,
a
n
∈
R
and
0
≤
a
0
=
a
n
≤
a
1
=
a
n
−
1
≤
⋯
≤
a
[
n
/
2
]
=
a
[
(
n
+
1
)
/
2
]
.
0 \leq a_0 = a_n \leq a_1 = a_{n-1 } \leq \cdots \leq a_{[n/2] }= a_{[(n+1)/2]}.
0
≤
a
0
=
a
n
≤
a
1
=
a
n
−
1
≤
⋯
≤
a
[
n
/2
]
=
a
[(
n
+
1
)
/2
]
.
Prove that if
f
∈
F
(
m
)
f \in F(m)
f
∈
F
(
m
)
and
g
∈
F
(
n
)
g \in F(n)
g
∈
F
(
n
)
, then
f
g
∈
F
(
m
+
n
)
.
fg \in F(m + n).
f
g
∈
F
(
m
+
n
)
.
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