MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam Team Selection Test
2019 Vietnam TST
P2
P2
Part of
2019 Vietnam TST
Problems
(1)
$P(x)$ has real roots
Source: VNTST 2019 P2
4/7/2019
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.
algebra
polynomial