MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
1991 Turkey Team Selection Test
3
Turkey TST 1991 - P3
Turkey TST 1991 - P3
Source:
March 13, 2011
function
algebra
polynomial
algebra proposed
Problem Statement
Let
f
f
f
be a function on defined on
∣
x
∣
<
1
|x|<1
∣
x
∣
<
1
such that
f
(
1
10
)
f\left (\tfrac1{10}\right )
f
(
10
1
)
is rational and
f
(
x
)
=
∑
i
=
1
∞
a
i
x
i
f(x)= \sum_{i=1}^{\infty} a_i x^i
f
(
x
)
=
∑
i
=
1
∞
a
i
x
i
where
a
i
∈
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}
a
i
∈
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
. Prove that
f
f
f
can be written as
f
(
x
)
=
p
(
x
)
q
(
x
)
f(x)= \frac{p(x)}{q(x)}
f
(
x
)
=
q
(
x
)
p
(
x
)
where
p
(
x
)
p(x)
p
(
x
)
and
q
(
x
)
q(x)
q
(
x
)
are polynomials with integer coefficients.
Back to Problems
View on AoPS