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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2016 Ukraine Team Selection Test
10
10
Part of
2016 Ukraine Team Selection Test
Problems
(1)
Root in (0; 2016)
Source: 2016 Ukraine TST
5/12/2018
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
)
.
algebra
polynomial
calculus
TST