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Vojtěch Jarník IMC
1995 VJIMC
Problem 2
Problem 2
Part of
1995 VJIMC
Problems
(2)
even function has extremum at 0
Source: VJIMC 1995 1.2
10/27/2021
Let
f
(
x
)
f(x)
f
(
x
)
be an even twice differentiable function such that
f
′
′
(
0
)
≠
0
f''(0)\ne0
f
′′
(
0
)
=
0
. Prove that
f
(
x
)
f(x)
f
(
x
)
has a local extremum at
x
=
0
x=0
x
=
0
.
function
polynomial in z+1/z
Source: VJIMC 1995 2.2
11/25/2021
Let
f
=
f
0
+
f
1
z
+
f
2
z
2
+
…
+
f
2
n
z
2
n
f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}
f
=
f
0
+
f
1
z
+
f
2
z
2
+
…
+
f
2
n
z
2
n
and
f
k
=
f
2
n
−
k
f_k=f_{2n-k}
f
k
=
f
2
n
−
k
for each
k
k
k
. Prove that
f
(
z
)
=
z
n
g
(
z
+
z
−
1
)
f(z)=z^ng(z+z^{-1})
f
(
z
)
=
z
n
g
(
z
+
z
−
1
)
, where
g
g
g
is a polynomial of degree
n
n
n
.
algebra
polynomial