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Soros Olympiad in Mathematics
VII Soros Olympiad 2000 - 01
11.2
3 parameter trinomial (VII Soros Olympiad 2000-01 R1 11.2)
3 parameter trinomial (VII Soros Olympiad 2000-01 R1 11.2)
Source:
July 29, 2021
parameterization
algebra
trinomial
Problem Statement
For all valid values of
a
,
b
a, b
a
,
b
, and
c
c
c
, solve the equation
a
(
x
−
b
)
(
x
−
c
)
(
a
−
b
)
(
a
−
c
)
+
b
(
x
−
c
)
(
x
−
a
)
(
b
−
c
)
(
b
−
a
)
+
c
(
x
−
a
)
(
x
−
b
)
(
c
−
a
)
(
c
−
b
)
=
x
2
\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2
(
a
−
b
)
(
a
−
c
)
a
(
x
−
b
)
(
x
−
c
)
+
(
b
−
c
)
(
b
−
a
)
b
(
x
−
c
)
(
x
−
a
)
+
(
c
−
a
)
(
c
−
b
)
c
(
x
−
a
)
(
x
−
b
)
=
x
2
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