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International Contests
Austrian-Polish
1991 Austrian-Polish Competition
4
4
Part of
1991 Austrian-Polish Competition
Problems
(1)
P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)
Source: Austrian Polish 1991 APMC
5/1/2020
Let
P
(
x
)
P(x)
P
(
x
)
be a real polynomial with
P
(
x
)
≥
0
P(x) \ge 0
P
(
x
)
≥
0
for
0
≤
x
≤
1
0 \le x \le 1
0
≤
x
≤
1
. Show that there exist polynomials
P
i
(
x
)
(
i
=
0
,
1
,
2
)
P_i (x) (i = 0, 1,2)
P
i
(
x
)
(
i
=
0
,
1
,
2
)
with
P
i
(
x
)
≥
0
P_i (x) \ge 0
P
i
(
x
)
≥
0
for all real x such that
P
(
x
)
=
P
0
(
x
)
+
x
P
1
(
x
)
(
1
−
x
)
P
2
(
x
)
P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)
P
(
x
)
=
P
0
(
x
)
+
x
P
1
(
x
)
(
1
−
x
)
P
2
(
x
)
.
polynomial
algebra