MathDB
Problems
Contests
International Contests
Baltic Way
1998 Baltic Way
6
6
Part of
1998 Baltic Way
Problems
(1)
The polynomial P is even if it is even for a,b
Source: Baltic Way 1998
1/11/2011
Let
P
P
P
be a polynomial of degree
6
6
6
and let
a
,
b
a,b
a
,
b
be real numbers such that
0
<
a
<
b
0<a<b
0
<
a
<
b
. Suppose that
P
(
a
)
=
P
(
−
a
)
,
P
(
b
)
=
P
(
−
b
)
,
P
′
(
0
)
=
0
P(a)=P(-a),P(b)=P(-b),P'(0)=0
P
(
a
)
=
P
(
−
a
)
,
P
(
b
)
=
P
(
−
b
)
,
P
′
(
0
)
=
0
. Prove that
P
(
x
)
=
P
(
−
x
)
P(x)=P(-x)
P
(
x
)
=
P
(
−
x
)
for all real
x
x
x
.
algebra
polynomial
algebra proposed