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India Contests
India National Olympiad
2000 India National Olympiad
3
Prove that they are equal
Prove that they are equal
Source: INMO 2000 Problem 3
October 10, 2005
algebra solved
algebra
Problem Statement
If
a
,
b
,
c
,
x
a,b,c,x
a
,
b
,
c
,
x
are real numbers such that
a
b
c
≠
0
abc \not= 0
ab
c
=
0
and
x
b
+
(
1
−
x
)
c
a
=
x
c
+
(
1
−
x
)
a
b
=
x
a
+
(
1
−
x
)
b
c
,
\frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c},
a
x
b
+
(
1
−
x
)
c
=
b
x
c
+
(
1
−
x
)
a
=
c
x
a
+
(
1
−
x
)
b
,
then prove that
a
=
b
=
c
a = b = c
a
=
b
=
c
.
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