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2016 Ukraine Team Selection Test
10
Root in (0; 2016)
Root in (0; 2016)
Source: 2016 Ukraine TST
May 12, 2018
algebra
polynomial
calculus
TST
Problem Statement
Let
a
1
,
…
,
a
n
a_1,\ldots, a_n
a
1
,
…
,
a
n
be real numbers. Define polynomials
f
,
g
f,g
f
,
g
by
f
(
x
)
=
∑
k
=
1
n
a
k
x
k
,
g
(
x
)
=
∑
k
=
1
n
a
k
2
k
−
1
x
k
.
f(x)=\sum_{k=1}^n a_kx^k,\ g(x)=\sum_{k=1}^n \frac{a_k}{2^k-1}x^k.
f
(
x
)
=
k
=
1
∑
n
a
k
x
k
,
g
(
x
)
=
k
=
1
∑
n
2
k
−
1
a
k
x
k
.
Assume that
g
(
2016
)
=
0
g(2016)=0
g
(
2016
)
=
0
. Prove that
f
(
x
)
f(x)
f
(
x
)
has a root in
(
0
;
2016
)
(0;2016)
(
0
;
2016
)
.
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