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National and Regional Contests
India Contests
Regional Mathematical Olympiad
2010 India Regional Mathematical Olympiad
2
Quadratic Polynomial
Quadratic Polynomial
Source:
December 5, 2010
quadratics
algebra
polynomial
Problem Statement
Let
P
1
(
x
)
=
a
x
2
−
b
x
−
c
P_1(x) = ax^2 - bx - c
P
1
(
x
)
=
a
x
2
−
b
x
−
c
,
P
2
(
x
)
=
b
x
2
−
c
x
−
a
P_2(x) = bx^2 - cx - a
P
2
(
x
)
=
b
x
2
−
c
x
−
a
,
P
3
(
x
)
=
c
x
2
−
a
x
−
b
P_3(x) = cx^2 - ax - b
P
3
(
x
)
=
c
x
2
−
a
x
−
b
be three quadratic polynomials. Suppose there exists a real number
α
\alpha
α
such that
P
1
(
α
)
=
P
2
(
α
)
=
P
3
(
α
)
P_1(\alpha) = P_2(\alpha) = P_3(\alpha)
P
1
(
α
)
=
P
2
(
α
)
=
P
3
(
α
)
. Prove that
a
=
b
=
c
a = b = c
a
=
b
=
c
.
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