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National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2019 Vietnam TST
P2
$P(x)$ has real roots
$P(x)$ has real roots
Source: VNTST 2019 P2
April 7, 2019
algebra
polynomial
Problem Statement
For each positive integer
n
n
n
, show that the polynomial:
P
n
(
x
)
=
∑
k
=
0
n
2
k
(
2
n
2
k
)
x
k
(
x
−
1
)
n
−
k
P_n(x)=\sum _{k=0}^n2^k\binom{2n}{2k}x^k(x-1)^{n-k}
P
n
(
x
)
=
k
=
0
∑
n
2
k
(
2
k
2
n
)
x
k
(
x
−
1
)
n
−
k
has
n
n
n
real roots.
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