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10
HMMT Team 2019/10: All roots on unit circle
HMMT Team 2019/10: All roots on unit circle
Source:
February 17, 2019
HMMT
algebra
Problem Statement
Prove that for all positive integers
n
n
n
, all complex roots
r
r
r
of the polynomial
P
(
x
)
=
(
2
n
)
x
2
n
+
(
2
n
−
1
)
x
2
n
−
1
+
⋯
+
(
n
+
1
)
x
n
+
1
+
n
x
n
+
(
n
+
1
)
x
n
−
1
+
⋯
+
(
2
n
−
1
)
x
+
2
n
P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n
P
(
x
)
=
(
2
n
)
x
2
n
+
(
2
n
−
1
)
x
2
n
−
1
+
⋯
+
(
n
+
1
)
x
n
+
1
+
n
x
n
+
(
n
+
1
)
x
n
−
1
+
⋯
+
(
2
n
−
1
)
x
+
2
n
lie on the unit circle (i.e.
∣
r
∣
=
1
|r| = 1
∣
r
∣
=
1
).
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