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Tuymaada Olympiad
2007 Tuymaada Olympiad
2
two polynomials of degree 100
two polynomials of degree 100
Source: Tuymaada 2007, Problem 2
July 15, 2007
algebra
polynomial
algebra proposed
Problem Statement
Two polynomials
f
(
x
)
=
a
100
x
100
+
a
99
x
99
+
⋯
+
a
1
x
+
a
0
f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}
f
(
x
)
=
a
100
x
100
+
a
99
x
99
+
⋯
+
a
1
x
+
a
0
and
g
(
x
)
=
b
100
x
100
+
b
99
x
99
+
⋯
+
b
1
x
+
b
0
g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}
g
(
x
)
=
b
100
x
100
+
b
99
x
99
+
⋯
+
b
1
x
+
b
0
of degree
100
100
100
differ from each other by a permutation of coefficients. It is known that
a
i
≠
b
i
a_{i}\ne b_{i}
a
i
=
b
i
for
i
=
0
,
1
,
2
,
…
,
100
i=0, 1, 2, \dots, 100
i
=
0
,
1
,
2
,
…
,
100
. Is it possible that
f
(
x
)
≥
g
(
x
)
f(x)\geq g(x)
f
(
x
)
≥
g
(
x
)
for all real
x
x
x
?
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