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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1996 Flanders Math Olympiad
4
4
Part of
1996 Flanders Math Olympiad
Problems
(1)
polynomial
Source: flanders '96
9/27/2005
Consider a real poylnomial
p
(
x
)
=
a
n
x
n
+
.
.
.
+
a
1
x
+
a
0
p(x)=a_nx^n+...+a_1x+a_0
p
(
x
)
=
a
n
x
n
+
...
+
a
1
x
+
a
0
. (a) If
deg
(
p
(
x
)
)
>
2
\deg(p(x))>2
de
g
(
p
(
x
))
>
2
prove that
deg
(
p
(
x
)
)
=
2
+
d
e
g
(
p
(
x
+
1
)
+
p
(
x
−
1
)
−
2
p
(
x
)
)
\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))
de
g
(
p
(
x
))
=
2
+
d
e
g
(
p
(
x
+
1
)
+
p
(
x
−
1
)
−
2
p
(
x
))
. (b) Let
p
(
x
)
p(x)
p
(
x
)
a polynomial for which there are real constants
r
,
s
r,s
r
,
s
so that for all real
x
x
x
we have
p
(
x
+
1
)
+
p
(
x
−
1
)
−
r
p
(
x
)
−
s
=
0
p(x+1)+p(x-1)-rp(x)-s=0
p
(
x
+
1
)
+
p
(
x
−
1
)
−
r
p
(
x
)
−
s
=
0
Prove
deg
(
p
(
x
)
)
≤
2
\deg(p(x))\le 2
de
g
(
p
(
x
))
≤
2
. (c) Show, in (b) that
s
=
0
s=0
s
=
0
implies
a
2
=
0
a_2=0
a
2
=
0
.
algebra
polynomial